Find the End Behavior y=(x^2-5)(x-1)^2(x-2)^3

$y = (x^{2} - 5) {(x - 1)}^{2} {(x - 2)}^{3}$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$(x^{2} - 5) ((x - 1) (x - 1)) {(x - 2)}^{3}$

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$(x^{2} - 5) (x (x - 1) - 1 (x - 1)) {(x - 2)}^{3}$

Apply the distributive property.

$(x^{2} - 5) (x \cdot x + x \cdot - 1 - 1 (x - 1)) {(x - 2)}^{3}$

Apply the distributive property.

$(x^{2} - 5) (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) {(x - 2)}^{3}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$(x^{2} - 5) (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) {(x - 2)}^{3}$

Move $- 1$ to the left of $x$ .

$(x^{2} - 5) (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) {(x - 2)}^{3}$

Rewrite $- 1 x$ as $- x$ .

$(x^{2} - 5) (x^{2} - x - 1 x - 1 \cdot - 1) {(x - 2)}^{3}$

Rewrite $- 1 x$ as $- x$ .

$(x^{2} - 5) (x^{2} - x - x - 1 \cdot - 1) {(x - 2)}^{3}$

Multiply $- 1$ by $- 1$ .

$(x^{2} - 5) (x^{2} - x - x + 1) {(x - 2)}^{3}$

Subtract $x$ from $- x$ .

$(x^{2} - 5) (x^{2} - 2 x + 1) {(x - 2)}^{3}$

Expand $(x^{2} - 5) (x^{2} - 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$(x^{2} x^{2} + x^{2} (- 2 x) + x^{2} \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

Simplify terms.

Simplify each term.

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

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

$(x^{2 + 2} + x^{2} (- 2 x) + x^{2} \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

Add $2$ and $2$ .

$(x^{4} + x^{2} (- 2 x) + x^{2} \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

Rewrite using the commutative property of multiplication.

$(x^{4} - 2 x^{2} x + x^{2} \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

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

Move $x$ .

$(x^{4} - 2 (x \cdot x^{2}) + x^{2} \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

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

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

$(x^{4} - 2 (x^{1} x^{2}) + x^{2} \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

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

$(x^{4} - 2 x^{1 + 2} + x^{2} \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

Add $1$ and $2$ .

$(x^{4} - 2 x^{3} + x^{2} \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

Multiply $x^{2}$ by $1$ .

$(x^{4} - 2 x^{3} + x^{2} - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

Multiply $- 2$ by $- 5$ .

$(x^{4} - 2 x^{3} + x^{2} - 5 x^{2} + 10 x - 5 \cdot 1) {(x - 2)}^{3}$

Multiply $- 5$ by $1$ .

$(x^{4} - 2 x^{3} + x^{2} - 5 x^{2} + 10 x - 5) {(x - 2)}^{3}$

Subtract $5 x^{2}$ from $x^{2}$ .

$(x^{4} - 2 x^{3} - 4 x^{2} + 10 x - 5) {(x - 2)}^{3}$

Use the Binomial Theorem.

$(x^{4} - 2 x^{3} - 4 x^{2} + 10 x - 5) (x^{3} + 3 x^{2} \cdot - 2 + 3 x {(- 2)}^{2} + {(- 2)}^{3})$

Simplify each term.

Multiply $- 2$ by $3$ .

$(x^{4} - 2 x^{3} - 4 x^{2} + 10 x - 5) (x^{3} - 6 x^{2} + 3 x {(- 2)}^{2} + {(- 2)}^{3})$

Raise $- 2$ to the power of $2$ .

$(x^{4} - 2 x^{3} - 4 x^{2} + 10 x - 5) (x^{3} - 6 x^{2} + 3 x \cdot 4 + {(- 2)}^{3})$

Multiply $4$ by $3$ .

$(x^{4} - 2 x^{3} - 4 x^{2} + 10 x - 5) (x^{3} - 6 x^{2} + 12 x + {(- 2)}^{3})$

Raise $- 2$ to the power of $3$ .

$(x^{4} - 2 x^{3} - 4 x^{2} + 10 x - 5) (x^{3} - 6 x^{2} + 12 x - 8)$

Expand $(x^{4} - 2 x^{3} - 4 x^{2} + 10 x - 5) (x^{3} - 6 x^{2} + 12 x - 8)$ by multiplying each term in the first expression by each term in the second expression.

$x^{4} x^{3} + x^{4} (- 6 x^{2}) + x^{4} (12 x) + x^{4} \cdot - 8 - 2 x^{3} x^{3} - 2 x^{3} (- 6 x^{2}) - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Simplify terms.

Simplify each term.

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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

$x^{4 + 3} + x^{4} (- 6 x^{2}) + x^{4} (12 x) + x^{4} \cdot - 8 - 2 x^{3} x^{3} - 2 x^{3} (- 6 x^{2}) - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Add $4$ and $3$ .

$x^{7} + x^{4} (- 6 x^{2}) + x^{4} (12 x) + x^{4} \cdot - 8 - 2 x^{3} x^{3} - 2 x^{3} (- 6 x^{2}) - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Rewrite using the commutative property of multiplication.

$x^{7} - 6 x^{4} x^{2} + x^{4} (12 x) + x^{4} \cdot - 8 - 2 x^{3} x^{3} - 2 x^{3} (- 6 x^{2}) - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

Move $x^{2}$ .

$x^{7} - 6 (x^{2} x^{4}) + x^{4} (12 x) + x^{4} \cdot - 8 - 2 x^{3} x^{3} - 2 x^{3} (- 6 x^{2}) - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

$x^{7} - 6 x^{2 + 4} + x^{4} (12 x) + x^{4} \cdot - 8 - 2 x^{3} x^{3} - 2 x^{3} (- 6 x^{2}) - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Add $2$ and $4$ .

$x^{7} - 6 x^{6} + x^{4} (12 x) + x^{4} \cdot - 8 - 2 x^{3} x^{3} - 2 x^{3} (- 6 x^{2}) - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Rewrite using the commutative property of multiplication.

$x^{7} - 6 x^{6} + 12 x^{4} x + x^{4} \cdot - 8 - 2 x^{3} x^{3} - 2 x^{3} (- 6 x^{2}) - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

Move $x$ .

$x^{7} - 6 x^{6} + 12 (x \cdot x^{4}) + x^{4} \cdot - 8 - 2 x^{3} x^{3} - 2 x^{3} (- 6 x^{2}) - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

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

$x^{7} - 6 x^{6} + 12 (x^{1} x^{4}) + x^{4} \cdot - 8 - 2 x^{3} x^{3} - 2 x^{3} (- 6 x^{2}) - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

$x^{7} - 6 x^{6} + 12 x^{1 + 4} + x^{4} \cdot - 8 - 2 x^{3} x^{3} - 2 x^{3} (- 6 x^{2}) - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Add $1$ and $4$ .

$x^{7} - 6 x^{6} + 12 x^{5} + x^{4} \cdot - 8 - 2 x^{3} x^{3} - 2 x^{3} (- 6 x^{2}) - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Move $- 8$ to the left of $x^{4}$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 \cdot x^{4} - 2 x^{3} x^{3} - 2 x^{3} (- 6 x^{2}) - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

Move $x^{3}$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 (x^{3} x^{3}) - 2 x^{3} (- 6 x^{2}) - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{3 + 3} - 2 x^{3} (- 6 x^{2}) - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Add $3$ and $3$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} - 2 x^{3} (- 6 x^{2}) - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Rewrite using the commutative property of multiplication.

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} - 2 \cdot - 6 x^{3} x^{2} - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

Move $x^{2}$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} - 2 \cdot - 6 (x^{2} x^{3}) - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} - 2 \cdot - 6 x^{2 + 3} - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Add $2$ and $3$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} - 2 \cdot - 6 x^{5} - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Multiply $- 2$ by $- 6$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 2 x^{3} (12 x) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Rewrite using the commutative property of multiplication.

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 2 \cdot 12 x^{3} x - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

Move $x$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 2 \cdot 12 (x \cdot x^{3}) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

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

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 2 \cdot 12 (x^{1} x^{3}) - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 2 \cdot 12 x^{1 + 3} - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Add $1$ and $3$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 2 \cdot 12 x^{4} - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Multiply $- 2$ by $12$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} - 2 x^{3} \cdot - 8 - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Multiply $- 8$ by $- 2$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{2} x^{3} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

Move $x^{3}$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 (x^{3} x^{2}) - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{3 + 2} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Add $3$ and $2$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} - 4 x^{2} (- 6 x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Rewrite using the commutative property of multiplication.

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} - 4 \cdot - 6 x^{2} x^{2} - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

Move $x^{2}$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} - 4 \cdot - 6 (x^{2} x^{2}) - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} - 4 \cdot - 6 x^{2 + 2} - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Add $2$ and $2$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} - 4 \cdot - 6 x^{4} - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Multiply $- 4$ by $- 6$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 4 x^{2} (12 x) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Rewrite using the commutative property of multiplication.

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 4 \cdot 12 x^{2} x - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

Move $x$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 4 \cdot 12 (x \cdot x^{2}) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

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

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 4 \cdot 12 (x^{1} x^{2}) - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 4 \cdot 12 x^{1 + 2} - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Add $1$ and $2$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 4 \cdot 12 x^{3} - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Multiply $- 4$ by $12$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} - 4 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Multiply $- 8$ by $- 4$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

Move $x^{3}$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 (x^{3} x) + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

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

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 (x^{3} x^{1}) + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{3 + 1} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Add $3$ and $1$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{4} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Rewrite using the commutative property of multiplication.

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{4} + 10 \cdot - 6 x \cdot x^{2} + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

Move $x^{2}$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{4} + 10 \cdot - 6 (x^{2} x) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

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

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{4} + 10 \cdot - 6 (x^{2} x^{1}) + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{4} + 10 \cdot - 6 x^{2 + 1} + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Add $2$ and $1$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{4} + 10 \cdot - 6 x^{3} + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Multiply $10$ by $- 6$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 10 x (12 x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Rewrite using the commutative property of multiplication.

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 10 \cdot 12 x \cdot x + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

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

Move $x$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 10 \cdot 12 (x \cdot x) + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Multiply $x$ by $x$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 10 \cdot 12 x^{2} + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Multiply $10$ by $12$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} + 10 x \cdot - 8 - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Multiply $- 8$ by $10$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} - 5 (- 6 x^{2}) - 5 (12 x) - 5 \cdot - 8$

Multiply $- 6$ by $- 5$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 5 (12 x) - 5 \cdot - 8$

Multiply $12$ by $- 5$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x - 5 \cdot - 8$

Multiply $- 5$ by $- 8$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Simplify by adding terms.

Combine the opposite terms in $x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$ .

Add $- 24 x^{4}$ and $24 x^{4}$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} + 16 x^{3} - 4 x^{5} + 0 - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Add $x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} + 16 x^{3} - 4 x^{5}$ and $0$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} + 16 x^{3} - 4 x^{5} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Subtract $2 x^{6}$ from $- 6 x^{6}$ .

$x^{7} - 8 x^{6} + 12 x^{5} - 8 x^{4} + 12 x^{5} + 16 x^{3} - 4 x^{5} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Add $12 x^{5}$ and $12 x^{5}$ .

$x^{7} - 8 x^{6} + 24 x^{5} - 8 x^{4} + 16 x^{3} - 4 x^{5} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Subtract $4 x^{5}$ from $24 x^{5}$ .

$x^{7} - 8 x^{6} + 20 x^{5} - 8 x^{4} + 16 x^{3} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Add $- 8 x^{4}$ and $10 x^{4}$ .

$x^{7} - 8 x^{6} + 20 x^{5} + 2 x^{4} + 16 x^{3} - 48 x^{3} + 32 x^{2} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Subtract $48 x^{3}$ from $16 x^{3}$ .

$x^{7} - 8 x^{6} + 20 x^{5} + 2 x^{4} - 32 x^{3} + 32 x^{2} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Subtract $60 x^{3}$ from $- 32 x^{3}$ .

$x^{7} - 8 x^{6} + 20 x^{5} + 2 x^{4} - 92 x^{3} + 32 x^{2} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Subtract $5 x^{3}$ from $- 92 x^{3}$ .

$x^{7} - 8 x^{6} + 20 x^{5} + 2 x^{4} - 97 x^{3} + 32 x^{2} + 120 x^{2} - 80 x + 30 x^{2} - 60 x + 40$

Add $32 x^{2}$ and $120 x^{2}$ .

$x^{7} - 8 x^{6} + 20 x^{5} + 2 x^{4} - 97 x^{3} + 152 x^{2} - 80 x + 30 x^{2} - 60 x + 40$

Add $152 x^{2}$ and $30 x^{2}$ .

$x^{7} - 8 x^{6} + 20 x^{5} + 2 x^{4} - 97 x^{3} + 182 x^{2} - 80 x - 60 x + 40$

Subtract $60 x$ from $- 80 x$ .

$x^{7} - 8 x^{6} + 20 x^{5} + 2 x^{4} - 97 x^{3} + 182 x^{2} - 140 x + 40$

The largest exponent is the degree of the polynomial.

$7$

Step 2

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

Step 3

Identify the leading coefficient.

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

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$(x^{2} - 5) ((x - 1) (x - 1)) {(x - 2)}^{3}$

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$(x^{2} - 5) (x (x - 1) - 1 (x - 1)) {(x - 2)}^{3}$

Apply the distributive property.

$(x^{2} - 5) (x \cdot x + x \cdot - 1 - 1 (x - 1)) {(x - 2)}^{3}$

Apply the distributive property.

$(x^{2} - 5) (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) {(x - 2)}^{3}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$(x^{2} - 5) (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) {(x - 2)}^{3}$

Move $- 1$ to the left of $x$ .

$(x^{2} - 5) (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) {(x - 2)}^{3}$

Rewrite $- 1 x$ as $- x$ .

$(x^{2} - 5) (x^{2} - x - 1 x - 1 \cdot - 1) {(x - 2)}^{3}$

Rewrite $- 1 x$ as $- x$ .

$(x^{2} - 5) (x^{2} - x - x - 1 \cdot - 1) {(x - 2)}^{3}$

Multiply $- 1$ by $- 1$ .

$(x^{2} - 5) (x^{2} - x - x + 1) {(x - 2)}^{3}$

Subtract $x$ from $- x$ .

$(x^{2} - 5) (x^{2} - 2 x + 1) {(x - 2)}^{3}$

Expand $(x^{2} - 5) (x^{2} - 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$(x^{2} x^{2} + x^{2} (- 2 x) + x^{2} \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

Simplify terms.

Simplify each term.

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

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

$(x^{2 + 2} + x^{2} (- 2 x) + x^{2} \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

Add $2$ and $2$ .

$(x^{4} + x^{2} (- 2 x) + x^{2} \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

Rewrite using the commutative property of multiplication.

$(x^{4} - 2 x^{2} x + x^{2} \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

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

Move $x$ .

$(x^{4} - 2 (x \cdot x^{2}) + x^{2} \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

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

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

$(x^{4} - 2 (x^{1} x^{2}) + x^{2} \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

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

$(x^{4} - 2 x^{1 + 2} + x^{2} \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

Add $1$ and $2$ .

$(x^{4} - 2 x^{3} + x^{2} \cdot 1 - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

Multiply $x^{2}$ by $1$ .

$(x^{4} - 2 x^{3} + x^{2} - 5 x^{2} - 5 (- 2 x) - 5 \cdot 1) {(x - 2)}^{3}$

Multiply $- 2$ by $- 5$ .

$(x^{4} - 2 x^{3} + x^{2} - 5 x^{2} + 10 x - 5 \cdot 1) {(x - 2)}^{3}$

Multiply $- 5$ by $1$ .

$(x^{4} - 2 x^{3} + x^{2} - 5 x^{2} + 10 x - 5) {(x - 2)}^{3}$

Subtract $5 x^{2}$ from $x^{2}$ .

$(x^{4} - 2 x^{3} - 4 x^{2} + 10 x - 5) {(x - 2)}^{3}$

Use the Binomial Theorem.

$(x^{4} - 2 x^{3} - 4 x^{2} + 10 x - 5) (x^{3} + 3 x^{2} \cdot - 2 + 3 x {(- 2)}^{2} + {(- 2)}^{3})$

Simplify each term.

Multiply $- 2$ by $3$ .

$(x^{4} - 2 x^{3} - 4 x^{2} + 10 x - 5) (x^{3} - 6 x^{2} + 3 x {(- 2)}^{2} + {(- 2)}^{3})$

Raise $- 2$ to the power of $2$ .

$(x^{4} - 2 x^{3} - 4 x^{2} + 10 x - 5) (x^{3} - 6 x^{2} + 3 x \cdot 4 + {(- 2)}^{3})$

Multiply $4$ by $3$ .

$(x^{4} - 2 x^{3} - 4 x^{2} + 10 x - 5) (x^{3} - 6 x^{2} + 12 x + {(- 2)}^{3})$

Raise $- 2$ to the power of $3$ .

$(x^{4} - 2 x^{3} - 4 x^{2} + 10 x - 5) (x^{3} - 6 x^{2} + 12 x - 8)$

Expand $(x^{4} - 2 x^{3} - 4 x^{2} + 10 x - 5) (x^{3} - 6 x^{2} + 12 x - 8)$ by multiplying each term in the first expression by each term in the second expression.

Simplify terms.

Simplify each term.

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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

Add $4$ and $3$ .

Rewrite using the commutative property of multiplication.

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 $4$ .

Move $- 8$ to the left of $x^{4}$ .

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 $- 2$ by $- 6$ .

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 $- 2$ by $12$ .

Multiply $- 8$ by $- 2$ .

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 $2$ .

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 $2$ .

Multiply $- 4$ by $- 6$ .

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 $- 4$ by $12$ .

Multiply $- 8$ by $- 4$ .

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

Move $x^{3}$ .

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

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

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

Add $3$ and $1$ .

Rewrite using the commutative property of multiplication.

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 $1$ .

Multiply $10$ by $- 6$ .

Rewrite using the commutative property of multiplication.

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

Move $x$ .

Multiply $x$ by $x$ .

Multiply $10$ by $12$ .

Multiply $- 8$ by $10$ .

Multiply $- 6$ by $- 5$ .

Multiply $12$ by $- 5$ .

Multiply $- 5$ by $- 8$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} - 24 x^{4} + 16 x^{3} - 4 x^{5} + 24 x^{4} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Simplify by adding terms.

Add $- 24 x^{4}$ and $24 x^{4}$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} + 16 x^{3} - 4 x^{5} + 0 - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Add $x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} + 16 x^{3} - 4 x^{5}$ and $0$ .

$x^{7} - 6 x^{6} + 12 x^{5} - 8 x^{4} - 2 x^{6} + 12 x^{5} + 16 x^{3} - 4 x^{5} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Subtract $2 x^{6}$ from $- 6 x^{6}$ .

$x^{7} - 8 x^{6} + 12 x^{5} - 8 x^{4} + 12 x^{5} + 16 x^{3} - 4 x^{5} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Add $12 x^{5}$ and $12 x^{5}$ .

$x^{7} - 8 x^{6} + 24 x^{5} - 8 x^{4} + 16 x^{3} - 4 x^{5} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Subtract $4 x^{5}$ from $24 x^{5}$ .

$x^{7} - 8 x^{6} + 20 x^{5} - 8 x^{4} + 16 x^{3} - 48 x^{3} + 32 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Add $- 8 x^{4}$ and $10 x^{4}$ .

$x^{7} - 8 x^{6} + 20 x^{5} + 2 x^{4} + 16 x^{3} - 48 x^{3} + 32 x^{2} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Subtract $48 x^{3}$ from $16 x^{3}$ .

$x^{7} - 8 x^{6} + 20 x^{5} + 2 x^{4} - 32 x^{3} + 32 x^{2} - 60 x^{3} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Subtract $60 x^{3}$ from $- 32 x^{3}$ .

$x^{7} - 8 x^{6} + 20 x^{5} + 2 x^{4} - 92 x^{3} + 32 x^{2} + 120 x^{2} - 80 x - 5 x^{3} + 30 x^{2} - 60 x + 40$

Subtract $5 x^{3}$ from $- 92 x^{3}$ .

$x^{7} - 8 x^{6} + 20 x^{5} + 2 x^{4} - 97 x^{3} + 32 x^{2} + 120 x^{2} - 80 x + 30 x^{2} - 60 x + 40$

Add $32 x^{2}$ and $120 x^{2}$ .

$x^{7} - 8 x^{6} + 20 x^{5} + 2 x^{4} - 97 x^{3} + 152 x^{2} - 80 x + 30 x^{2} - 60 x + 40$

Add $152 x^{2}$ and $30 x^{2}$ .

$x^{7} - 8 x^{6} + 20 x^{5} + 2 x^{4} - 97 x^{3} + 182 x^{2} - 80 x - 60 x + 40$

Subtract $60 x$ from $- 80 x$ .

$x^{7} - 8 x^{6} + 20 x^{5} + 2 x^{4} - 97 x^{3} + 182 x^{2} - 140 x + 40$

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

$x^{7}$

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

$1$

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.

Falls to the left and rises to the right

Step 7

Find the End Behavior y=(x^2-5)(x-1)^2(x-2)^3

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