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

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

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

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

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $3$ to the left of $x$ .

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

Multiply $3$ by $3$ .

$(x - 2) (x^{2} + 3 x + 3 x + 9) {(x + 1)}^{3}$

Add $3 x$ and $3 x$ .

$(x - 2) (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Expand $(x - 2) (x^{2} + 6 x + 9)$ by multiplying each term in the first expression by each term in the second expression.

$(x \cdot x^{2} + x (6 x) + x \cdot 9 - 2 x^{2} - 2 (6 x) - 2 \cdot 9) {(x + 1)}^{3}$

Simplify terms.

Simplify each term.

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

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

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

$(x^{1} x^{2} + x (6 x) + x \cdot 9 - 2 x^{2} - 2 (6 x) - 2 \cdot 9) {(x + 1)}^{3}$

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

$(x^{1 + 2} + x (6 x) + x \cdot 9 - 2 x^{2} - 2 (6 x) - 2 \cdot 9) {(x + 1)}^{3}$

Add $1$ and $2$ .

$(x^{3} + x (6 x) + x \cdot 9 - 2 x^{2} - 2 (6 x) - 2 \cdot 9) {(x + 1)}^{3}$

Rewrite using the commutative property of multiplication.

$(x^{3} + 6 x \cdot x + x \cdot 9 - 2 x^{2} - 2 (6 x) - 2 \cdot 9) {(x + 1)}^{3}$

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

Move $x$ .

$(x^{3} + 6 (x \cdot x) + x \cdot 9 - 2 x^{2} - 2 (6 x) - 2 \cdot 9) {(x + 1)}^{3}$

Multiply $x$ by $x$ .

$(x^{3} + 6 x^{2} + x \cdot 9 - 2 x^{2} - 2 (6 x) - 2 \cdot 9) {(x + 1)}^{3}$

Move $9$ to the left of $x$ .

$(x^{3} + 6 x^{2} + 9 \cdot x - 2 x^{2} - 2 (6 x) - 2 \cdot 9) {(x + 1)}^{3}$

Multiply $6$ by $- 2$ .

$(x^{3} + 6 x^{2} + 9 x - 2 x^{2} - 12 x - 2 \cdot 9) {(x + 1)}^{3}$

Multiply $- 2$ by $9$ .

$(x^{3} + 6 x^{2} + 9 x - 2 x^{2} - 12 x - 18) {(x + 1)}^{3}$

Simplify by adding terms.

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

$(x^{3} + 4 x^{2} + 9 x - 12 x - 18) {(x + 1)}^{3}$

Subtract $12 x$ from $9 x$ .

$(x^{3} + 4 x^{2} - 3 x - 18) {(x + 1)}^{3}$

Use the Binomial Theorem.

$(x^{3} + 4 x^{2} - 3 x - 18) (x^{3} + 3 x^{2} \cdot 1 + 3 x \cdot 1^{2} + 1^{3})$

Simplify each term.

Multiply $3$ by $1$ .

$(x^{3} + 4 x^{2} - 3 x - 18) (x^{3} + 3 x^{2} + 3 x \cdot 1^{2} + 1^{3})$

One to any power is one.

$(x^{3} + 4 x^{2} - 3 x - 18) (x^{3} + 3 x^{2} + 3 x \cdot 1 + 1^{3})$

Multiply $3$ by $1$ .

$(x^{3} + 4 x^{2} - 3 x - 18) (x^{3} + 3 x^{2} + 3 x + 1^{3})$

One to any power is one.

$(x^{3} + 4 x^{2} - 3 x - 18) (x^{3} + 3 x^{2} + 3 x + 1)$

Expand $(x^{3} + 4 x^{2} - 3 x - 18) (x^{3} + 3 x^{2} + 3 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$x^{3} x^{3} + x^{3} (3 x^{2}) + x^{3} (3 x) + x^{3} \cdot 1 + 4 x^{2} x^{3} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x \cdot x^{3} - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Simplify terms.

Combine the opposite terms in $x^{3} x^{3} + x^{3} (3 x^{2}) + x^{3} (3 x) + x^{3} \cdot 1 + 4 x^{2} x^{3} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x \cdot x^{3} - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$ .

Reorder the factors in the terms $x^{3} (3 x)$ and $- 3 x \cdot x^{3}$ .

$x^{3} x^{3} + x^{3} (3 x^{2}) + 3 x^{3} x + x^{3} \cdot 1 + 4 x^{2} x^{3} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x^{3} x - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

$x^{3} x^{3} + x^{3} (3 x^{2}) + x^{3} \cdot 1 + 4 x^{2} x^{3} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 + 0 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Add $x^{3} x^{3} + x^{3} (3 x^{2}) + x^{3} \cdot 1 + 4 x^{2} x^{3} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1$ and $0$ .

$x^{3} x^{3} + x^{3} (3 x^{2}) + x^{3} \cdot 1 + 4 x^{2} x^{3} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Simplify each term.

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

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

$x^{3 + 3} + x^{3} (3 x^{2}) + x^{3} \cdot 1 + 4 x^{2} x^{3} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Add $3$ and $3$ .

$x^{6} + x^{3} (3 x^{2}) + x^{3} \cdot 1 + 4 x^{2} x^{3} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Rewrite using the commutative property of multiplication.

$x^{6} + 3 x^{3} x^{2} + x^{3} \cdot 1 + 4 x^{2} x^{3} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

Move $x^{2}$ .

$x^{6} + 3 (x^{2} x^{3}) + x^{3} \cdot 1 + 4 x^{2} x^{3} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

$x^{6} + 3 x^{2 + 3} + x^{3} \cdot 1 + 4 x^{2} x^{3} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Add $2$ and $3$ .

$x^{6} + 3 x^{5} + x^{3} \cdot 1 + 4 x^{2} x^{3} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $x^{3}$ by $1$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{2} x^{3} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

Move $x^{3}$ .

$x^{6} + 3 x^{5} + x^{3} + 4 (x^{3} x^{2}) + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

$x^{6} + 3 x^{5} + x^{3} + 4 x^{3 + 2} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Add $3$ and $2$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Rewrite using the commutative property of multiplication.

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 4 \cdot 3 x^{2} x^{2} + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

Move $x^{2}$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 4 \cdot 3 (x^{2} x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 4 \cdot 3 x^{2 + 2} + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Add $2$ and $2$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 4 \cdot 3 x^{4} + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $4$ by $3$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Rewrite using the commutative property of multiplication.

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 4 \cdot 3 x^{2} x + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

Move $x$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 4 \cdot 3 (x \cdot x^{2}) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

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

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 4 \cdot 3 (x^{1} x^{2}) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 4 \cdot 3 x^{1 + 2} + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Add $1$ and $2$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 4 \cdot 3 x^{3} + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $4$ by $3$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $4$ by $1$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Rewrite using the commutative property of multiplication.

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 3 \cdot 3 x \cdot x^{2} - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

Move $x^{2}$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 3 \cdot 3 (x^{2} x) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

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

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 3 \cdot 3 (x^{2} x^{1}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 3 \cdot 3 x^{2 + 1} - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Add $2$ and $1$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 3 \cdot 3 x^{3} - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $- 3$ by $3$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Rewrite using the commutative property of multiplication.

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 3 \cdot 3 x \cdot x - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

Move $x$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 3 \cdot 3 (x \cdot x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $x$ by $x$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 3 \cdot 3 x^{2} - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $- 3$ by $3$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 9 x^{2} - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $- 3$ by $1$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 9 x^{2} - 3 x - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $3$ by $- 18$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 9 x^{2} - 3 x - 18 x^{3} - 54 x^{2} - 18 (3 x) - 18 \cdot 1$

Multiply $3$ by $- 18$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 9 x^{2} - 3 x - 18 x^{3} - 54 x^{2} - 54 x - 18 \cdot 1$

Multiply $- 18$ by $1$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 9 x^{2} - 3 x - 18 x^{3} - 54 x^{2} - 54 x - 18$

Simplify by adding terms.

Add $3 x^{5}$ and $4 x^{5}$ .

$x^{6} + 7 x^{5} + x^{3} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 9 x^{2} - 3 x - 18 x^{3} - 54 x^{2} - 54 x - 18$

Add $x^{3}$ and $12 x^{3}$ .

$x^{6} + 7 x^{5} + 12 x^{4} + 13 x^{3} + 4 x^{2} - 9 x^{3} - 9 x^{2} - 3 x - 18 x^{3} - 54 x^{2} - 54 x - 18$

Subtract $9 x^{3}$ from $13 x^{3}$ .

$x^{6} + 7 x^{5} + 12 x^{4} + 4 x^{3} + 4 x^{2} - 9 x^{2} - 3 x - 18 x^{3} - 54 x^{2} - 54 x - 18$

Subtract $18 x^{3}$ from $4 x^{3}$ .

$x^{6} + 7 x^{5} + 12 x^{4} - 14 x^{3} + 4 x^{2} - 9 x^{2} - 3 x - 54 x^{2} - 54 x - 18$

Subtract $9 x^{2}$ from $4 x^{2}$ .

$x^{6} + 7 x^{5} + 12 x^{4} - 14 x^{3} - 5 x^{2} - 3 x - 54 x^{2} - 54 x - 18$

Subtract $54 x^{2}$ from $- 5 x^{2}$ .

$x^{6} + 7 x^{5} + 12 x^{4} - 14 x^{3} - 59 x^{2} - 3 x - 54 x - 18$

Subtract $54 x$ from $- 3 x$ .

$x^{6} + 7 x^{5} + 12 x^{4} - 14 x^{3} - 59 x^{2} - 57 x - 18$

The largest exponent is the degree of the polynomial.

$6$

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 ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

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

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $3$ to the left of $x$ .

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

Multiply $3$ by $3$ .

$(x - 2) (x^{2} + 3 x + 3 x + 9) {(x + 1)}^{3}$

Add $3 x$ and $3 x$ .

$(x - 2) (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Expand $(x - 2) (x^{2} + 6 x + 9)$ by multiplying each term in the first expression by each term in the second expression.

$(x \cdot x^{2} + x (6 x) + x \cdot 9 - 2 x^{2} - 2 (6 x) - 2 \cdot 9) {(x + 1)}^{3}$

Simplify terms.

Simplify each term.

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

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

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

$(x^{1} x^{2} + x (6 x) + x \cdot 9 - 2 x^{2} - 2 (6 x) - 2 \cdot 9) {(x + 1)}^{3}$

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

$(x^{1 + 2} + x (6 x) + x \cdot 9 - 2 x^{2} - 2 (6 x) - 2 \cdot 9) {(x + 1)}^{3}$

Add $1$ and $2$ .

$(x^{3} + x (6 x) + x \cdot 9 - 2 x^{2} - 2 (6 x) - 2 \cdot 9) {(x + 1)}^{3}$

Rewrite using the commutative property of multiplication.

$(x^{3} + 6 x \cdot x + x \cdot 9 - 2 x^{2} - 2 (6 x) - 2 \cdot 9) {(x + 1)}^{3}$

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

Move $x$ .

$(x^{3} + 6 (x \cdot x) + x \cdot 9 - 2 x^{2} - 2 (6 x) - 2 \cdot 9) {(x + 1)}^{3}$

Multiply $x$ by $x$ .

$(x^{3} + 6 x^{2} + x \cdot 9 - 2 x^{2} - 2 (6 x) - 2 \cdot 9) {(x + 1)}^{3}$

Move $9$ to the left of $x$ .

$(x^{3} + 6 x^{2} + 9 \cdot x - 2 x^{2} - 2 (6 x) - 2 \cdot 9) {(x + 1)}^{3}$

Multiply $6$ by $- 2$ .

$(x^{3} + 6 x^{2} + 9 x - 2 x^{2} - 12 x - 2 \cdot 9) {(x + 1)}^{3}$

Multiply $- 2$ by $9$ .

$(x^{3} + 6 x^{2} + 9 x - 2 x^{2} - 12 x - 18) {(x + 1)}^{3}$

Simplify by adding terms.

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

$(x^{3} + 4 x^{2} + 9 x - 12 x - 18) {(x + 1)}^{3}$

Subtract $12 x$ from $9 x$ .

$(x^{3} + 4 x^{2} - 3 x - 18) {(x + 1)}^{3}$

Use the Binomial Theorem.

$(x^{3} + 4 x^{2} - 3 x - 18) (x^{3} + 3 x^{2} \cdot 1 + 3 x \cdot 1^{2} + 1^{3})$

Simplify each term.

Multiply $3$ by $1$ .

$(x^{3} + 4 x^{2} - 3 x - 18) (x^{3} + 3 x^{2} + 3 x \cdot 1^{2} + 1^{3})$

One to any power is one.

$(x^{3} + 4 x^{2} - 3 x - 18) (x^{3} + 3 x^{2} + 3 x \cdot 1 + 1^{3})$

Multiply $3$ by $1$ .

$(x^{3} + 4 x^{2} - 3 x - 18) (x^{3} + 3 x^{2} + 3 x + 1^{3})$

One to any power is one.

$(x^{3} + 4 x^{2} - 3 x - 18) (x^{3} + 3 x^{2} + 3 x + 1)$

Expand $(x^{3} + 4 x^{2} - 3 x - 18) (x^{3} + 3 x^{2} + 3 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

Simplify terms.

Reorder the factors in the terms $x^{3} (3 x)$ and $- 3 x \cdot x^{3}$ .

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

Add $x^{3} x^{3} + x^{3} (3 x^{2}) + x^{3} \cdot 1 + 4 x^{2} x^{3} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1$ and $0$ .

Simplify each term.

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

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.

$x^{6} + 3 x^{2 + 3} + x^{3} \cdot 1 + 4 x^{2} x^{3} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Add $2$ and $3$ .

$x^{6} + 3 x^{5} + x^{3} \cdot 1 + 4 x^{2} x^{3} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $x^{3}$ by $1$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{2} x^{3} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

Move $x^{3}$ .

$x^{6} + 3 x^{5} + x^{3} + 4 (x^{3} x^{2}) + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

$x^{6} + 3 x^{5} + x^{3} + 4 x^{3 + 2} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Add $3$ and $2$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 4 x^{2} (3 x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Rewrite using the commutative property of multiplication.

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 4 \cdot 3 x^{2} x^{2} + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

Move $x^{2}$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 4 \cdot 3 (x^{2} x^{2}) + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 4 \cdot 3 x^{2 + 2} + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Add $2$ and $2$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 4 \cdot 3 x^{4} + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $4$ by $3$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 4 x^{2} (3 x) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Rewrite using the commutative property of multiplication.

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 4 \cdot 3 x^{2} x + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

Move $x$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 4 \cdot 3 (x \cdot x^{2}) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

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

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 4 \cdot 3 (x^{1} x^{2}) + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 4 \cdot 3 x^{1 + 2} + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Add $1$ and $2$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 4 \cdot 3 x^{3} + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $4$ by $3$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} \cdot 1 - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $4$ by $1$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 3 x (3 x^{2}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Rewrite using the commutative property of multiplication.

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 3 \cdot 3 x \cdot x^{2} - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

Move $x^{2}$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 3 \cdot 3 (x^{2} x) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

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

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 3 \cdot 3 (x^{2} x^{1}) - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 3 \cdot 3 x^{2 + 1} - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Add $2$ and $1$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 3 \cdot 3 x^{3} - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $- 3$ by $3$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 3 x (3 x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Rewrite using the commutative property of multiplication.

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 3 \cdot 3 x \cdot x - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

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

Move $x$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 3 \cdot 3 (x \cdot x) - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $x$ by $x$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 3 \cdot 3 x^{2} - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $- 3$ by $3$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 9 x^{2} - 3 x \cdot 1 - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $- 3$ by $1$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 9 x^{2} - 3 x - 18 x^{3} - 18 (3 x^{2}) - 18 (3 x) - 18 \cdot 1$

Multiply $3$ by $- 18$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 9 x^{2} - 3 x - 18 x^{3} - 54 x^{2} - 18 (3 x) - 18 \cdot 1$

Multiply $3$ by $- 18$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 9 x^{2} - 3 x - 18 x^{3} - 54 x^{2} - 54 x - 18 \cdot 1$

Multiply $- 18$ by $1$ .

$x^{6} + 3 x^{5} + x^{3} + 4 x^{5} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 9 x^{2} - 3 x - 18 x^{3} - 54 x^{2} - 54 x - 18$

Simplify by adding terms.

Add $3 x^{5}$ and $4 x^{5}$ .

$x^{6} + 7 x^{5} + x^{3} + 12 x^{4} + 12 x^{3} + 4 x^{2} - 9 x^{3} - 9 x^{2} - 3 x - 18 x^{3} - 54 x^{2} - 54 x - 18$

Add $x^{3}$ and $12 x^{3}$ .

$x^{6} + 7 x^{5} + 12 x^{4} + 13 x^{3} + 4 x^{2} - 9 x^{3} - 9 x^{2} - 3 x - 18 x^{3} - 54 x^{2} - 54 x - 18$

Subtract $9 x^{3}$ from $13 x^{3}$ .

$x^{6} + 7 x^{5} + 12 x^{4} + 4 x^{3} + 4 x^{2} - 9 x^{2} - 3 x - 18 x^{3} - 54 x^{2} - 54 x - 18$

Subtract $18 x^{3}$ from $4 x^{3}$ .

$x^{6} + 7 x^{5} + 12 x^{4} - 14 x^{3} + 4 x^{2} - 9 x^{2} - 3 x - 54 x^{2} - 54 x - 18$

Subtract $9 x^{2}$ from $4 x^{2}$ .

$x^{6} + 7 x^{5} + 12 x^{4} - 14 x^{3} - 5 x^{2} - 3 x - 54 x^{2} - 54 x - 18$

Subtract $54 x^{2}$ from $- 5 x^{2}$ .

$x^{6} + 7 x^{5} + 12 x^{4} - 14 x^{3} - 59 x^{2} - 3 x - 54 x - 18$

Subtract $54 x$ from $- 3 x$ .

$x^{6} + 7 x^{5} + 12 x^{4} - 14 x^{3} - 59 x^{2} - 57 x - 18$

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

$x^{6}$

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.

Rises to the left and rises to the right

Step 7

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

Related Questions

Support LCMGCF.com by disabling your adblocker.