Find the End Behavior f(x)=11x(x+7)^2(x+1)^3(x-2)^2

$f (x) = 11 x {(x + 7)}^{2} {(x + 1)}^{3} {(x - 2)}^{2}$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Rewrite ${(x + 7)}^{2}$ as $(x + 7) (x + 7)$ .

$11 x ((x + 7) (x + 7)) {(x + 1)}^{3} {(x - 2)}^{2}$

Expand $(x + 7) (x + 7)$ using the FOIL Method.

Apply the distributive property.

$11 x (x (x + 7) + 7 (x + 7)) {(x + 1)}^{3} {(x - 2)}^{2}$

Apply the distributive property.

$11 x (x \cdot x + x \cdot 7 + 7 (x + 7)) {(x + 1)}^{3} {(x - 2)}^{2}$

Apply the distributive property.

$11 x (x \cdot x + x \cdot 7 + 7 x + 7 \cdot 7) {(x + 1)}^{3} {(x - 2)}^{2}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$11 x (x^{2} + x \cdot 7 + 7 x + 7 \cdot 7) {(x + 1)}^{3} {(x - 2)}^{2}$

Move $7$ to the left of $x$ .

$11 x (x^{2} + 7 \cdot x + 7 x + 7 \cdot 7) {(x + 1)}^{3} {(x - 2)}^{2}$

Multiply $7$ by $7$ .

$11 x (x^{2} + 7 x + 7 x + 49) {(x + 1)}^{3} {(x - 2)}^{2}$

Add $7 x$ and $7 x$ .

$11 x (x^{2} + 14 x + 49) {(x + 1)}^{3} {(x - 2)}^{2}$

Apply the distributive property.

$(11 x \cdot x^{2} + 11 x (14 x) + 11 x \cdot 49) {(x + 1)}^{3} {(x - 2)}^{2}$

Simplify.

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(11 (x^{2} x) + 11 x (14 x) + 11 x \cdot 49) {(x + 1)}^{3} {(x - 2)}^{2}$

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$(11 (x^{2} x^{1}) + 11 x (14 x) + 11 x \cdot 49) {(x + 1)}^{3} {(x - 2)}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(11 x^{2 + 1} + 11 x (14 x) + 11 x \cdot 49) {(x + 1)}^{3} {(x - 2)}^{2}$

Add $2$ and $1$ .

$(11 x^{3} + 11 x (14 x) + 11 x \cdot 49) {(x + 1)}^{3} {(x - 2)}^{2}$

Rewrite using the commutative property of multiplication.

$(11 x^{3} + 11 \cdot 14 x \cdot x + 11 x \cdot 49) {(x + 1)}^{3} {(x - 2)}^{2}$

Multiply $49$ by $11$ .

$(11 x^{3} + 11 \cdot 14 x \cdot x + 539 x) {(x + 1)}^{3} {(x - 2)}^{2}$

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$(11 x^{3} + 11 \cdot 14 (x \cdot x) + 539 x) {(x + 1)}^{3} {(x - 2)}^{2}$

Multiply $x$ by $x$ .

$(11 x^{3} + 11 \cdot 14 x^{2} + 539 x) {(x + 1)}^{3} {(x - 2)}^{2}$

Multiply $11$ by $14$ .

$(11 x^{3} + 154 x^{2} + 539 x) {(x + 1)}^{3} {(x - 2)}^{2}$

Use the Binomial Theorem.

$(11 x^{3} + 154 x^{2} + 539 x) (x^{3} + 3 x^{2} \cdot 1 + 3 x \cdot 1^{2} + 1^{3}) {(x - 2)}^{2}$

Simplify each term.

Multiply $3$ by $1$ .

$(11 x^{3} + 154 x^{2} + 539 x) (x^{3} + 3 x^{2} + 3 x \cdot 1^{2} + 1^{3}) {(x - 2)}^{2}$

One to any power is one.

$(11 x^{3} + 154 x^{2} + 539 x) (x^{3} + 3 x^{2} + 3 x \cdot 1 + 1^{3}) {(x - 2)}^{2}$

Multiply $3$ by $1$ .

$(11 x^{3} + 154 x^{2} + 539 x) (x^{3} + 3 x^{2} + 3 x + 1^{3}) {(x - 2)}^{2}$

One to any power is one.

$(11 x^{3} + 154 x^{2} + 539 x) (x^{3} + 3 x^{2} + 3 x + 1) {(x - 2)}^{2}$

Expand $(11 x^{3} + 154 x^{2} + 539 x) (x^{3} + 3 x^{2} + 3 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$(11 x^{3} x^{3} + 11 x^{3} (3 x^{2}) + 11 x^{3} (3 x) + 11 x^{3} \cdot 1 + 154 x^{2} x^{3} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Simplify terms.

Simplify each term.

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$(11 (x^{3} x^{3}) + 11 x^{3} (3 x^{2}) + 11 x^{3} (3 x) + 11 x^{3} \cdot 1 + 154 x^{2} x^{3} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(11 x^{3 + 3} + 11 x^{3} (3 x^{2}) + 11 x^{3} (3 x) + 11 x^{3} \cdot 1 + 154 x^{2} x^{3} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Add $3$ and $3$ .

$(11 x^{6} + 11 x^{3} (3 x^{2}) + 11 x^{3} (3 x) + 11 x^{3} \cdot 1 + 154 x^{2} x^{3} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Rewrite using the commutative property of multiplication.

$(11 x^{6} + 11 \cdot 3 x^{3} x^{2} + 11 x^{3} (3 x) + 11 x^{3} \cdot 1 + 154 x^{2} x^{3} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(11 x^{6} + 11 \cdot 3 (x^{2} x^{3}) + 11 x^{3} (3 x) + 11 x^{3} \cdot 1 + 154 x^{2} x^{3} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(11 x^{6} + 11 \cdot 3 x^{2 + 3} + 11 x^{3} (3 x) + 11 x^{3} \cdot 1 + 154 x^{2} x^{3} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Add $2$ and $3$ .

$(11 x^{6} + 11 \cdot 3 x^{5} + 11 x^{3} (3 x) + 11 x^{3} \cdot 1 + 154 x^{2} x^{3} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $11$ by $3$ .

$(11 x^{6} + 33 x^{5} + 11 x^{3} (3 x) + 11 x^{3} \cdot 1 + 154 x^{2} x^{3} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Rewrite using the commutative property of multiplication.

$(11 x^{6} + 33 x^{5} + 11 \cdot 3 x^{3} x + 11 x^{3} \cdot 1 + 154 x^{2} x^{3} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x^{3}$ by $x$ by adding the exponents.

Move $x$ .

$(11 x^{6} + 33 x^{5} + 11 \cdot 3 (x \cdot x^{3}) + 11 x^{3} \cdot 1 + 154 x^{2} x^{3} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x$ by $x^{3}$ .

Raise $x$ to the power of $1$ .

$(11 x^{6} + 33 x^{5} + 11 \cdot 3 (x^{1} x^{3}) + 11 x^{3} \cdot 1 + 154 x^{2} x^{3} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(11 x^{6} + 33 x^{5} + 11 \cdot 3 x^{1 + 3} + 11 x^{3} \cdot 1 + 154 x^{2} x^{3} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Add $1$ and $3$ .

$(11 x^{6} + 33 x^{5} + 11 \cdot 3 x^{4} + 11 x^{3} \cdot 1 + 154 x^{2} x^{3} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $11$ by $3$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} \cdot 1 + 154 x^{2} x^{3} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $11$ by $1$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{2} x^{3} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 (x^{3} x^{2}) + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{3 + 2} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Add $3$ and $2$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Rewrite using the commutative property of multiplication.

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 154 \cdot 3 x^{2} x^{2} + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 154 \cdot 3 (x^{2} x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 154 \cdot 3 x^{2 + 2} + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Add $2$ and $2$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 154 \cdot 3 x^{4} + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $154$ by $3$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Rewrite using the commutative property of multiplication.

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 154 \cdot 3 x^{2} x + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 154 \cdot 3 (x \cdot x^{2}) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 154 \cdot 3 (x^{1} x^{2}) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 154 \cdot 3 x^{1 + 2} + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Add $1$ and $2$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 154 \cdot 3 x^{3} + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $154$ by $3$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $154$ by $1$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 (x^{3} x) + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x^{3}$ by $x$ .

Raise $x$ to the power of $1$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 (x^{3} x^{1}) + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{3 + 1} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Add $3$ and $1$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Rewrite using the commutative property of multiplication.

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 539 \cdot 3 x \cdot x^{2} + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 539 \cdot 3 (x^{2} x) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 539 \cdot 3 (x^{2} x^{1}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 539 \cdot 3 x^{2 + 1} + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Add $2$ and $1$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 539 \cdot 3 x^{3} + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $539$ by $3$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 1617 x^{3} + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Rewrite using the commutative property of multiplication.

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 1617 x^{3} + 539 \cdot 3 x \cdot x + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 1617 x^{3} + 539 \cdot 3 (x \cdot x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x$ by $x$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 1617 x^{3} + 539 \cdot 3 x^{2} + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $539$ by $3$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 1617 x^{3} + 1617 x^{2} + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $539$ by $1$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 1617 x^{3} + 1617 x^{2} + 539 x) {(x - 2)}^{2}$

Simplify by adding terms.

Add $33 x^{5}$ and $154 x^{5}$ .

$(11 x^{6} + 187 x^{5} + 33 x^{4} + 11 x^{3} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 1617 x^{3} + 1617 x^{2} + 539 x) {(x - 2)}^{2}$

Add $33 x^{4}$ and $462 x^{4}$ .

$(11 x^{6} + 187 x^{5} + 495 x^{4} + 11 x^{3} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 1617 x^{3} + 1617 x^{2} + 539 x) {(x - 2)}^{2}$

Add $495 x^{4}$ and $539 x^{4}$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 11 x^{3} + 462 x^{3} + 154 x^{2} + 1617 x^{3} + 1617 x^{2} + 539 x) {(x - 2)}^{2}$

Add $11 x^{3}$ and $462 x^{3}$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 473 x^{3} + 154 x^{2} + 1617 x^{3} + 1617 x^{2} + 539 x) {(x - 2)}^{2}$

Add $473 x^{3}$ and $1617 x^{3}$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 154 x^{2} + 1617 x^{2} + 539 x) {(x - 2)}^{2}$

Add $154 x^{2}$ and $1617 x^{2}$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) {(x - 2)}^{2}$

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) ((x - 2) (x - 2))$

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) (x (x - 2) - 2 (x - 2))$

Apply the distributive property.

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) (x \cdot x + x \cdot - 2 - 2 (x - 2))$

Apply the distributive property.

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2)$

Move $- 2$ to the left of $x$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2)$

Multiply $- 2$ by $- 2$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) (x^{2} - 2 x - 2 x + 4)$

Subtract $2 x$ from $- 2 x$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) (x^{2} - 4 x + 4)$

Expand $(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) (x^{2} - 4 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

$11 x^{6} x^{2} + 11 x^{6} (- 4 x) + 11 x^{6} \cdot 4 + 187 x^{5} x^{2} + 187 x^{5} (- 4 x) + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Simplify terms.

Simplify each term.

Multiply $x^{6}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$11 (x^{2} x^{6}) + 11 x^{6} (- 4 x) + 11 x^{6} \cdot 4 + 187 x^{5} x^{2} + 187 x^{5} (- 4 x) + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$11 x^{2 + 6} + 11 x^{6} (- 4 x) + 11 x^{6} \cdot 4 + 187 x^{5} x^{2} + 187 x^{5} (- 4 x) + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Add $2$ and $6$ .

$11 x^{8} + 11 x^{6} (- 4 x) + 11 x^{6} \cdot 4 + 187 x^{5} x^{2} + 187 x^{5} (- 4 x) + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Rewrite using the commutative property of multiplication.

$11 x^{8} + 11 \cdot - 4 x^{6} x + 11 x^{6} \cdot 4 + 187 x^{5} x^{2} + 187 x^{5} (- 4 x) + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $x^{6}$ by $x$ by adding the exponents.

Move $x$ .

$11 x^{8} + 11 \cdot - 4 (x \cdot x^{6}) + 11 x^{6} \cdot 4 + 187 x^{5} x^{2} + 187 x^{5} (- 4 x) + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $x$ by $x^{6}$ .

Raise $x$ to the power of $1$ .

$11 x^{8} + 11 \cdot - 4 (x^{1} x^{6}) + 11 x^{6} \cdot 4 + 187 x^{5} x^{2} + 187 x^{5} (- 4 x) + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$11 x^{8} + 11 \cdot - 4 x^{1 + 6} + 11 x^{6} \cdot 4 + 187 x^{5} x^{2} + 187 x^{5} (- 4 x) + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Add $1$ and $6$ .

$11 x^{8} + 11 \cdot - 4 x^{7} + 11 x^{6} \cdot 4 + 187 x^{5} x^{2} + 187 x^{5} (- 4 x) + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $11$ by $- 4$ .

$11 x^{8} - 44 x^{7} + 11 x^{6} \cdot 4 + 187 x^{5} x^{2} + 187 x^{5} (- 4 x) + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $4$ by $11$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{5} x^{2} + 187 x^{5} (- 4 x) + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 (x^{2} x^{5}) + 187 x^{5} (- 4 x) + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{2 + 5} + 187 x^{5} (- 4 x) + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Add $2$ and $5$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} + 187 x^{5} (- 4 x) + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Rewrite using the commutative property of multiplication.

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} + 187 \cdot - 4 x^{5} x + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $x^{5}$ by $x$ by adding the exponents.

Move $x$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} + 187 \cdot - 4 (x \cdot x^{5}) + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $x$ by $x^{5}$ .

Raise $x$ to the power of $1$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} + 187 \cdot - 4 (x^{1} x^{5}) + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} + 187 \cdot - 4 x^{1 + 5} + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Add $1$ and $5$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} + 187 \cdot - 4 x^{6} + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $187$ by $- 4$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 187 x^{5} \cdot 4 + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $4$ by $187$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{4} x^{2} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 (x^{2} x^{4}) + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{2 + 4} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Add $2$ and $4$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} + 1034 x^{4} (- 4 x) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Rewrite using the commutative property of multiplication.

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} + 1034 \cdot - 4 x^{4} x + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $x^{4}$ by $x$ by adding the exponents.

Move $x$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} + 1034 \cdot - 4 (x \cdot x^{4}) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $x$ by $x^{4}$ .

Raise $x$ to the power of $1$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} + 1034 \cdot - 4 (x^{1} x^{4}) + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} + 1034 \cdot - 4 x^{1 + 4} + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Add $1$ and $4$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} + 1034 \cdot - 4 x^{5} + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $1034$ by $- 4$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 1034 x^{4} \cdot 4 + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $4$ by $1034$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{3} x^{2} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 (x^{2} x^{3}) + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{2 + 3} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Add $2$ and $3$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} + 2090 x^{3} (- 4 x) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Rewrite using the commutative property of multiplication.

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} + 2090 \cdot - 4 x^{3} x + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $x^{3}$ by $x$ by adding the exponents.

Move $x$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} + 2090 \cdot - 4 (x \cdot x^{3}) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $x$ by $x^{3}$ .

Raise $x$ to the power of $1$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} + 2090 \cdot - 4 (x^{1} x^{3}) + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} + 2090 \cdot - 4 x^{1 + 3} + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Add $1$ and $3$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} + 2090 \cdot - 4 x^{4} + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $2090$ by $- 4$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 2090 x^{3} \cdot 4 + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $4$ by $2090$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{2} x^{2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 (x^{2} x^{2}) + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{2 + 2} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Add $2$ and $2$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} + 1771 x^{2} (- 4 x) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Rewrite using the commutative property of multiplication.

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} + 1771 \cdot - 4 x^{2} x + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} + 1771 \cdot - 4 (x \cdot x^{2}) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} + 1771 \cdot - 4 (x^{1} x^{2}) + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} + 1771 \cdot - 4 x^{1 + 2} + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Add $1$ and $2$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} + 1771 \cdot - 4 x^{3} + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $1771$ by $- 4$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} - 7084 x^{3} + 1771 x^{2} \cdot 4 + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $4$ by $1771$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} - 7084 x^{3} + 7084 x^{2} + 539 x \cdot x^{2} + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} - 7084 x^{3} + 7084 x^{2} + 539 (x^{2} x) + 539 x (- 4 x) + 539 x \cdot 4$

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} - 7084 x^{3} + 7084 x^{2} + 539 x^{2 + 1} + 539 x (- 4 x) + 539 x \cdot 4$

Add $2$ and $1$ .

$11 x^{8} - 44 x^{7} + 44 x^{6} + 187 x^{7} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} - 7084 x^{3} + 7084 x^{2} + 539 x^{3} + 539 x (- 4 x) + 539 x \cdot 4$

Rewrite using the commutative property of multiplication.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

Multiply $x$ by $x$ .

Multiply $539$ by $- 4$ .

Multiply $4$ by $539$ .

Simplify by adding terms.

Add $- 44 x^{7}$ and $187 x^{7}$ .

$11 x^{8} + 143 x^{7} + 44 x^{6} - 748 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} - 7084 x^{3} + 7084 x^{2} + 539 x^{3} - 2156 x^{2} + 2156 x$

Subtract $748 x^{6}$ from $44 x^{6}$ .

$11 x^{8} + 143 x^{7} - 704 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} - 7084 x^{3} + 7084 x^{2} + 539 x^{3} - 2156 x^{2} + 2156 x$

Add $- 704 x^{6}$ and $1034 x^{6}$ .

$11 x^{8} + 143 x^{7} + 330 x^{6} + 748 x^{5} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} - 7084 x^{3} + 7084 x^{2} + 539 x^{3} - 2156 x^{2} + 2156 x$

Subtract $4136 x^{5}$ from $748 x^{5}$ .

$11 x^{8} + 143 x^{7} + 330 x^{6} - 3388 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} - 7084 x^{3} + 7084 x^{2} + 539 x^{3} - 2156 x^{2} + 2156 x$

Add $- 3388 x^{5}$ and $2090 x^{5}$ .

$11 x^{8} + 143 x^{7} + 330 x^{6} - 1298 x^{5} + 4136 x^{4} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} - 7084 x^{3} + 7084 x^{2} + 539 x^{3} - 2156 x^{2} + 2156 x$

Subtract $8360 x^{4}$ from $4136 x^{4}$ .

$11 x^{8} + 143 x^{7} + 330 x^{6} - 1298 x^{5} - 4224 x^{4} + 8360 x^{3} + 1771 x^{4} - 7084 x^{3} + 7084 x^{2} + 539 x^{3} - 2156 x^{2} + 2156 x$

Add $- 4224 x^{4}$ and $1771 x^{4}$ .

$11 x^{8} + 143 x^{7} + 330 x^{6} - 1298 x^{5} - 2453 x^{4} + 8360 x^{3} - 7084 x^{3} + 7084 x^{2} + 539 x^{3} - 2156 x^{2} + 2156 x$

Subtract $7084 x^{3}$ from $8360 x^{3}$ .

$11 x^{8} + 143 x^{7} + 330 x^{6} - 1298 x^{5} - 2453 x^{4} + 1276 x^{3} + 7084 x^{2} + 539 x^{3} - 2156 x^{2} + 2156 x$

Add $1276 x^{3}$ and $539 x^{3}$ .

$11 x^{8} + 143 x^{7} + 330 x^{6} - 1298 x^{5} - 2453 x^{4} + 1815 x^{3} + 7084 x^{2} - 2156 x^{2} + 2156 x$

Subtract $2156 x^{2}$ from $7084 x^{2}$ .

$11 x^{8} + 143 x^{7} + 330 x^{6} - 1298 x^{5} - 2453 x^{4} + 1815 x^{3} + 4928 x^{2} + 2156 x$

The largest exponent is the degree of the polynomial.

$8$

Step 2

Since the degree is even, the ends of the function will point in the same direction.

Even

Step 3

Identify the leading coefficient.

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

Rewrite ${(x + 7)}^{2}$ as $(x + 7) (x + 7)$ .

$11 x ((x + 7) (x + 7)) {(x + 1)}^{3} {(x - 2)}^{2}$

Expand $(x + 7) (x + 7)$ using the FOIL Method.

Apply the distributive property.

$11 x (x (x + 7) + 7 (x + 7)) {(x + 1)}^{3} {(x - 2)}^{2}$

Apply the distributive property.

$11 x (x \cdot x + x \cdot 7 + 7 (x + 7)) {(x + 1)}^{3} {(x - 2)}^{2}$

Apply the distributive property.

$11 x (x \cdot x + x \cdot 7 + 7 x + 7 \cdot 7) {(x + 1)}^{3} {(x - 2)}^{2}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$11 x (x^{2} + x \cdot 7 + 7 x + 7 \cdot 7) {(x + 1)}^{3} {(x - 2)}^{2}$

Move $7$ to the left of $x$ .

$11 x (x^{2} + 7 \cdot x + 7 x + 7 \cdot 7) {(x + 1)}^{3} {(x - 2)}^{2}$

Multiply $7$ by $7$ .

$11 x (x^{2} + 7 x + 7 x + 49) {(x + 1)}^{3} {(x - 2)}^{2}$

Add $7 x$ and $7 x$ .

$11 x (x^{2} + 14 x + 49) {(x + 1)}^{3} {(x - 2)}^{2}$

Apply the distributive property.

$(11 x \cdot x^{2} + 11 x (14 x) + 11 x \cdot 49) {(x + 1)}^{3} {(x - 2)}^{2}$

Simplify.

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(11 (x^{2} x) + 11 x (14 x) + 11 x \cdot 49) {(x + 1)}^{3} {(x - 2)}^{2}$

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$(11 (x^{2} x^{1}) + 11 x (14 x) + 11 x \cdot 49) {(x + 1)}^{3} {(x - 2)}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(11 x^{2 + 1} + 11 x (14 x) + 11 x \cdot 49) {(x + 1)}^{3} {(x - 2)}^{2}$

Add $2$ and $1$ .

$(11 x^{3} + 11 x (14 x) + 11 x \cdot 49) {(x + 1)}^{3} {(x - 2)}^{2}$

Rewrite using the commutative property of multiplication.

$(11 x^{3} + 11 \cdot 14 x \cdot x + 11 x \cdot 49) {(x + 1)}^{3} {(x - 2)}^{2}$

Multiply $49$ by $11$ .

$(11 x^{3} + 11 \cdot 14 x \cdot x + 539 x) {(x + 1)}^{3} {(x - 2)}^{2}$

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$(11 x^{3} + 11 \cdot 14 (x \cdot x) + 539 x) {(x + 1)}^{3} {(x - 2)}^{2}$

Multiply $x$ by $x$ .

$(11 x^{3} + 11 \cdot 14 x^{2} + 539 x) {(x + 1)}^{3} {(x - 2)}^{2}$

Multiply $11$ by $14$ .

$(11 x^{3} + 154 x^{2} + 539 x) {(x + 1)}^{3} {(x - 2)}^{2}$

Use the Binomial Theorem.

$(11 x^{3} + 154 x^{2} + 539 x) (x^{3} + 3 x^{2} \cdot 1 + 3 x \cdot 1^{2} + 1^{3}) {(x - 2)}^{2}$

Simplify each term.

Multiply $3$ by $1$ .

$(11 x^{3} + 154 x^{2} + 539 x) (x^{3} + 3 x^{2} + 3 x \cdot 1^{2} + 1^{3}) {(x - 2)}^{2}$

One to any power is one.

$(11 x^{3} + 154 x^{2} + 539 x) (x^{3} + 3 x^{2} + 3 x \cdot 1 + 1^{3}) {(x - 2)}^{2}$

Multiply $3$ by $1$ .

$(11 x^{3} + 154 x^{2} + 539 x) (x^{3} + 3 x^{2} + 3 x + 1^{3}) {(x - 2)}^{2}$

One to any power is one.

$(11 x^{3} + 154 x^{2} + 539 x) (x^{3} + 3 x^{2} + 3 x + 1) {(x - 2)}^{2}$

Expand $(11 x^{3} + 154 x^{2} + 539 x) (x^{3} + 3 x^{2} + 3 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

Simplify terms.

Simplify each term.

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Add $3$ and $3$ .

Rewrite using the commutative property of multiplication.

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Add $2$ and $3$ .

Multiply $11$ by $3$ .

Rewrite using the commutative property of multiplication.

Multiply $x^{3}$ by $x$ by adding the exponents.

Move $x$ .

Multiply $x$ by $x^{3}$ .

Raise $x$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Add $1$ and $3$ .

Multiply $11$ by $3$ .

Multiply $11$ by $1$ .

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Add $3$ and $2$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 154 x^{2} (3 x^{2}) + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Rewrite using the commutative property of multiplication.

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Add $2$ and $2$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 154 \cdot 3 x^{4} + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $154$ by $3$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 154 x^{2} (3 x) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Rewrite using the commutative property of multiplication.

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 154 \cdot 3 x^{2} x + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 154 \cdot 3 (x^{1} x^{2}) + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 154 \cdot 3 x^{1 + 2} + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Add $1$ and $2$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 154 \cdot 3 x^{3} + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $154$ by $3$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} \cdot 1 + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $154$ by $1$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x \cdot x^{3} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 (x^{3} x) + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x^{3}$ by $x$ .

Raise $x$ to the power of $1$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 (x^{3} x^{1}) + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{3 + 1} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Add $3$ and $1$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 539 x (3 x^{2}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Rewrite using the commutative property of multiplication.

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 539 \cdot 3 x \cdot x^{2} + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 539 \cdot 3 (x^{2} x) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 539 \cdot 3 (x^{2} x^{1}) + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 539 \cdot 3 x^{2 + 1} + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Add $2$ and $1$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 539 \cdot 3 x^{3} + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $539$ by $3$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 1617 x^{3} + 539 x (3 x) + 539 x \cdot 1) {(x - 2)}^{2}$

Rewrite using the commutative property of multiplication.

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 1617 x^{3} + 539 \cdot 3 x \cdot x + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 1617 x^{3} + 539 \cdot 3 (x \cdot x) + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $x$ by $x$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 1617 x^{3} + 539 \cdot 3 x^{2} + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $539$ by $3$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 1617 x^{3} + 1617 x^{2} + 539 x \cdot 1) {(x - 2)}^{2}$

Multiply $539$ by $1$ .

$(11 x^{6} + 33 x^{5} + 33 x^{4} + 11 x^{3} + 154 x^{5} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 1617 x^{3} + 1617 x^{2} + 539 x) {(x - 2)}^{2}$

Simplify by adding terms.

Add $33 x^{5}$ and $154 x^{5}$ .

$(11 x^{6} + 187 x^{5} + 33 x^{4} + 11 x^{3} + 462 x^{4} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 1617 x^{3} + 1617 x^{2} + 539 x) {(x - 2)}^{2}$

Add $33 x^{4}$ and $462 x^{4}$ .

$(11 x^{6} + 187 x^{5} + 495 x^{4} + 11 x^{3} + 462 x^{3} + 154 x^{2} + 539 x^{4} + 1617 x^{3} + 1617 x^{2} + 539 x) {(x - 2)}^{2}$

Add $495 x^{4}$ and $539 x^{4}$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 11 x^{3} + 462 x^{3} + 154 x^{2} + 1617 x^{3} + 1617 x^{2} + 539 x) {(x - 2)}^{2}$

Add $11 x^{3}$ and $462 x^{3}$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 473 x^{3} + 154 x^{2} + 1617 x^{3} + 1617 x^{2} + 539 x) {(x - 2)}^{2}$

Add $473 x^{3}$ and $1617 x^{3}$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 154 x^{2} + 1617 x^{2} + 539 x) {(x - 2)}^{2}$

Add $154 x^{2}$ and $1617 x^{2}$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) {(x - 2)}^{2}$

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) ((x - 2) (x - 2))$

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) (x (x - 2) - 2 (x - 2))$

Apply the distributive property.

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) (x \cdot x + x \cdot - 2 - 2 (x - 2))$

Apply the distributive property.

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2)$

Move $- 2$ to the left of $x$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2)$

Multiply $- 2$ by $- 2$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) (x^{2} - 2 x - 2 x + 4)$

Subtract $2 x$ from $- 2 x$ .

$(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) (x^{2} - 4 x + 4)$

Expand $(11 x^{6} + 187 x^{5} + 1034 x^{4} + 2090 x^{3} + 1771 x^{2} + 539 x) (x^{2} - 4 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

Simplify terms.

Simplify each term.

Multiply $x^{6}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Add $2$ and $6$ .

Rewrite using the commutative property of multiplication.

Multiply $x^{6}$ by $x$ by adding the exponents.

Move $x$ .

Multiply $x$ by $x^{6}$ .

Raise $x$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Add $1$ and $6$ .

Multiply $11$ by $- 4$ .

Multiply $4$ by $11$ .

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Add $2$ and $5$ .

Rewrite using the commutative property of multiplication.

Multiply $x^{5}$ by $x$ by adding the exponents.

Move $x$ .

Multiply $x$ by $x^{5}$ .

Raise $x$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Add $1$ and $5$ .

Multiply $187$ by $- 4$ .

Multiply $4$ by $187$ .

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Add $2$ and $4$ .

Rewrite using the commutative property of multiplication.

Multiply $x^{4}$ by $x$ by adding the exponents.

Move $x$ .

Multiply $x$ by $x^{4}$ .

Raise $x$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Add $1$ and $4$ .

Multiply $1034$ by $- 4$ .

Multiply $4$ by $1034$ .

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Add $2$ and $3$ .

Rewrite using the commutative property of multiplication.

Multiply $x^{3}$ by $x$ by adding the exponents.

Move $x$ .

Multiply $x$ by $x^{3}$ .

Raise $x$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Add $1$ and $3$ .

Multiply $2090$ by $- 4$ .

Multiply $4$ by $2090$ .

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Add $2$ and $2$ .

Rewrite using the commutative property of multiplication.

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Add $1$ and $2$ .

Multiply $1771$ by $- 4$ .

Multiply $4$ by $1771$ .

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Add $2$ and $1$ .

Rewrite using the commutative property of multiplication.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

Multiply $x$ by $x$ .

Multiply $539$ by $- 4$ .

Multiply $4$ by $539$ .

Simplify by adding terms.

Add $- 44 x^{7}$ and $187 x^{7}$ .

Subtract $748 x^{6}$ from $44 x^{6}$ .

$11 x^{8} + 143 x^{7} - 704 x^{6} + 748 x^{5} + 1034 x^{6} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} - 7084 x^{3} + 7084 x^{2} + 539 x^{3} - 2156 x^{2} + 2156 x$

Add $- 704 x^{6}$ and $1034 x^{6}$ .

$11 x^{8} + 143 x^{7} + 330 x^{6} + 748 x^{5} - 4136 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} - 7084 x^{3} + 7084 x^{2} + 539 x^{3} - 2156 x^{2} + 2156 x$

Subtract $4136 x^{5}$ from $748 x^{5}$ .

$11 x^{8} + 143 x^{7} + 330 x^{6} - 3388 x^{5} + 4136 x^{4} + 2090 x^{5} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} - 7084 x^{3} + 7084 x^{2} + 539 x^{3} - 2156 x^{2} + 2156 x$

Add $- 3388 x^{5}$ and $2090 x^{5}$ .

$11 x^{8} + 143 x^{7} + 330 x^{6} - 1298 x^{5} + 4136 x^{4} - 8360 x^{4} + 8360 x^{3} + 1771 x^{4} - 7084 x^{3} + 7084 x^{2} + 539 x^{3} - 2156 x^{2} + 2156 x$

Subtract $8360 x^{4}$ from $4136 x^{4}$ .

$11 x^{8} + 143 x^{7} + 330 x^{6} - 1298 x^{5} - 4224 x^{4} + 8360 x^{3} + 1771 x^{4} - 7084 x^{3} + 7084 x^{2} + 539 x^{3} - 2156 x^{2} + 2156 x$

Add $- 4224 x^{4}$ and $1771 x^{4}$ .

$11 x^{8} + 143 x^{7} + 330 x^{6} - 1298 x^{5} - 2453 x^{4} + 8360 x^{3} - 7084 x^{3} + 7084 x^{2} + 539 x^{3} - 2156 x^{2} + 2156 x$

Subtract $7084 x^{3}$ from $8360 x^{3}$ .

$11 x^{8} + 143 x^{7} + 330 x^{6} - 1298 x^{5} - 2453 x^{4} + 1276 x^{3} + 7084 x^{2} + 539 x^{3} - 2156 x^{2} + 2156 x$

Add $1276 x^{3}$ and $539 x^{3}$ .

$11 x^{8} + 143 x^{7} + 330 x^{6} - 1298 x^{5} - 2453 x^{4} + 1815 x^{3} + 7084 x^{2} - 2156 x^{2} + 2156 x$

Subtract $2156 x^{2}$ from $7084 x^{2}$ .

$11 x^{8} + 143 x^{7} + 330 x^{6} - 1298 x^{5} - 2453 x^{4} + 1815 x^{3} + 4928 x^{2} + 2156 x$

The leading term in a polynomial is the term with the highest degree.

$11 x^{8}$

The leading coefficient in a polynomial is the coefficient of the leading term.

$11$

Step 4

Since the leading coefficient is positive, the graph rises to the right.

Positive

Step 5

Use the degree of the function, as well as the sign of the leading coefficient to determine the behavior.

1. Even and Positive: Rises to the left and rises to the right.

2. Even and Negative: Falls to the left and falls to the right.

3. Odd and Positive: Falls to the left and rises to the right.

4. Odd and Negative: Rises to the left and falls to the right

Step 6

Determine the behavior.

Rises to the left and rises to the right

Step 7

Find the End Behavior f(x)=11x(x+7)^2(x+1)^3(x-2)^2

Related Questions

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