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

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

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $x$ by $1$ .

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

Multiply $1$ by $1$ .

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

Add $x$ and $x$ .

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

Apply the distributive property.

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

Simplify.

Multiply $2$ by $5$ .

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

Multiply $5$ by $1$ .

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

Use the Binomial Theorem.

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

Simplify each term.

Multiply $- 2$ by $3$ .

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

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

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

Multiply $4$ by $3$ .

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

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

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

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

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

Simplify terms.

Simplify each term.

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

Move $x^{3}$ .

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

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

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

Add $3$ and $2$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Multiply $5$ by $- 6$ .

$5 x^{5} - 30 x^{4} + 5 x^{2} (12 x) + 5 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Rewrite using the commutative property of multiplication.

$5 x^{5} - 30 x^{4} + 5 \cdot 12 x^{2} x + 5 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

Move $x$ .

$5 x^{5} - 30 x^{4} + 5 \cdot 12 (x \cdot x^{2}) + 5 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

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

$5 x^{5} - 30 x^{4} + 5 \cdot 12 (x^{1} x^{2}) + 5 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

$5 x^{5} - 30 x^{4} + 5 \cdot 12 x^{1 + 2} + 5 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Add $1$ and $2$ .

$5 x^{5} - 30 x^{4} + 5 \cdot 12 x^{3} + 5 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Multiply $5$ by $12$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} + 5 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Multiply $- 8$ by $5$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

Move $x^{3}$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 (x^{3} x) + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

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

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 (x^{3} x^{1}) + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{3 + 1} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Add $3$ and $1$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Rewrite using the commutative property of multiplication.

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} + 10 \cdot - 6 x \cdot x^{2} + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

Move $x^{2}$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} + 10 \cdot - 6 (x^{2} x) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

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

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} + 10 \cdot - 6 (x^{2} x^{1}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} + 10 \cdot - 6 x^{2 + 1} + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Add $2$ and $1$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} + 10 \cdot - 6 x^{3} + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Multiply $10$ by $- 6$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Rewrite using the commutative property of multiplication.

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 10 \cdot 12 x \cdot x + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

Move $x$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 10 \cdot 12 (x \cdot x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Multiply $x$ by $x$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 10 \cdot 12 x^{2} + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Multiply $10$ by $12$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Multiply $- 8$ by $10$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Multiply $- 6$ by $5$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x + 5 x^{3} - 30 x^{2} + 5 (12 x) + 5 \cdot - 8$

Multiply $12$ by $5$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x + 5 x^{3} - 30 x^{2} + 60 x + 5 \cdot - 8$

Multiply $5$ by $- 8$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x + 5 x^{3} - 30 x^{2} + 60 x - 40$

Simplify by adding terms.

Combine the opposite terms in $5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x + 5 x^{3} - 30 x^{2} + 60 x - 40$ .

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

$5 x^{5} - 30 x^{4} - 40 x^{2} + 10 x^{4} + 0 + 120 x^{2} - 80 x + 5 x^{3} - 30 x^{2} + 60 x - 40$

Add $5 x^{5} - 30 x^{4} - 40 x^{2} + 10 x^{4}$ and $0$ .

$5 x^{5} - 30 x^{4} - 40 x^{2} + 10 x^{4} + 120 x^{2} - 80 x + 5 x^{3} - 30 x^{2} + 60 x - 40$

Add $- 30 x^{4}$ and $10 x^{4}$ .

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

Add $- 40 x^{2}$ and $120 x^{2}$ .

$5 x^{5} - 20 x^{4} + 80 x^{2} - 80 x + 5 x^{3} - 30 x^{2} + 60 x - 40$

Subtract $30 x^{2}$ from $80 x^{2}$ .

$5 x^{5} - 20 x^{4} + 50 x^{2} - 80 x + 5 x^{3} + 60 x - 40$

Add $- 80 x$ and $60 x$ .

$5 x^{5} - 20 x^{4} + 50 x^{2} + 5 x^{3} - 20 x - 40$

The largest exponent is the degree of the polynomial.

$5$

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 $(x + 1) (x + 1)$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $x$ by $1$ .

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

Multiply $1$ by $1$ .

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

Add $x$ and $x$ .

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

Apply the distributive property.

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

Simplify.

Multiply $2$ by $5$ .

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

Multiply $5$ by $1$ .

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

Use the Binomial Theorem.

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

Simplify each term.

Multiply $- 2$ by $3$ .

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

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

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

Multiply $4$ by $3$ .

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

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

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

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

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

Simplify terms.

Simplify each term.

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

Move $x^{3}$ .

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

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

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

Add $3$ and $2$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Multiply $5$ by $- 6$ .

$5 x^{5} - 30 x^{4} + 5 x^{2} (12 x) + 5 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Rewrite using the commutative property of multiplication.

$5 x^{5} - 30 x^{4} + 5 \cdot 12 x^{2} x + 5 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

Move $x$ .

$5 x^{5} - 30 x^{4} + 5 \cdot 12 (x \cdot x^{2}) + 5 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

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

$5 x^{5} - 30 x^{4} + 5 \cdot 12 (x^{1} x^{2}) + 5 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

$5 x^{5} - 30 x^{4} + 5 \cdot 12 x^{1 + 2} + 5 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Add $1$ and $2$ .

$5 x^{5} - 30 x^{4} + 5 \cdot 12 x^{3} + 5 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Multiply $5$ by $12$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} + 5 x^{2} \cdot - 8 + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Multiply $- 8$ by $5$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x \cdot x^{3} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

Move $x^{3}$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 (x^{3} x) + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

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

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 (x^{3} x^{1}) + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{3 + 1} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Add $3$ and $1$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} + 10 x (- 6 x^{2}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Rewrite using the commutative property of multiplication.

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} + 10 \cdot - 6 x \cdot x^{2} + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

Move $x^{2}$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} + 10 \cdot - 6 (x^{2} x) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

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

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} + 10 \cdot - 6 (x^{2} x^{1}) + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} + 10 \cdot - 6 x^{2 + 1} + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Add $2$ and $1$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} + 10 \cdot - 6 x^{3} + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Multiply $10$ by $- 6$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 10 x (12 x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Rewrite using the commutative property of multiplication.

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 10 \cdot 12 x \cdot x + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

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

Move $x$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 10 \cdot 12 (x \cdot x) + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Multiply $x$ by $x$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 10 \cdot 12 x^{2} + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Multiply $10$ by $12$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} + 10 x \cdot - 8 + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Multiply $- 8$ by $10$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x + 5 x^{3} + 5 (- 6 x^{2}) + 5 (12 x) + 5 \cdot - 8$

Multiply $- 6$ by $5$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x + 5 x^{3} - 30 x^{2} + 5 (12 x) + 5 \cdot - 8$

Multiply $12$ by $5$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x + 5 x^{3} - 30 x^{2} + 60 x + 5 \cdot - 8$

Multiply $5$ by $- 8$ .

$5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x + 5 x^{3} - 30 x^{2} + 60 x - 40$

Simplify by adding terms.

Combine the opposite terms in $5 x^{5} - 30 x^{4} + 60 x^{3} - 40 x^{2} + 10 x^{4} - 60 x^{3} + 120 x^{2} - 80 x + 5 x^{3} - 30 x^{2} + 60 x - 40$ .

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

$5 x^{5} - 30 x^{4} - 40 x^{2} + 10 x^{4} + 0 + 120 x^{2} - 80 x + 5 x^{3} - 30 x^{2} + 60 x - 40$

Add $5 x^{5} - 30 x^{4} - 40 x^{2} + 10 x^{4}$ and $0$ .

$5 x^{5} - 30 x^{4} - 40 x^{2} + 10 x^{4} + 120 x^{2} - 80 x + 5 x^{3} - 30 x^{2} + 60 x - 40$

Add $- 30 x^{4}$ and $10 x^{4}$ .

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

Add $- 40 x^{2}$ and $120 x^{2}$ .

$5 x^{5} - 20 x^{4} + 80 x^{2} - 80 x + 5 x^{3} - 30 x^{2} + 60 x - 40$

Subtract $30 x^{2}$ from $80 x^{2}$ .

$5 x^{5} - 20 x^{4} + 50 x^{2} - 80 x + 5 x^{3} + 60 x - 40$

Add $- 80 x$ and $60 x$ .

$5 x^{5} - 20 x^{4} + 50 x^{2} + 5 x^{3} - 20 x - 40$

Move $50 x^{2}$ .

$5 x^{5} - 20 x^{4} + 5 x^{3} + 50 x^{2} - 20 x - 40$

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

$5 x^{5}$

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

$5$

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 P(x)=5(x+1)^2(x-2)^3

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