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

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

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Simplify by multiplying through.

Apply the distributive property.

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

Simplify the expression.

Multiply $- 1$ by $3$ .

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

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

$(- x - 3) ((x + 1) (x + 1)) (2 x - 5)$

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

Apply the distributive property.

$(- x - 3) (x (x + 1) + 1 (x + 1)) (2 x - 5)$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $x$ by $1$ .

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

Multiply $1$ by $1$ .

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

Add $x$ and $x$ .

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

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

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

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $- 1$ by $1$ .

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

Multiply $2$ by $- 3$ .

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

Multiply $- 3$ by $1$ .

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

Simplify by adding terms.

Subtract $3 x^{2}$ from $- 2 x^{2}$ .

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

Subtract $6 x$ from $- x$ .

$(- x^{3} - 5 x^{2} - 7 x - 3) (2 x - 5)$

Expand $(- x^{3} - 5 x^{2} - 7 x - 3) (2 x - 5)$ by multiplying each term in the first expression by each term in the second expression.

$- x^{3} (2 x) - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$- 1 \cdot 2 x^{3} x - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

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

Move $x$ .

$- 1 \cdot 2 (x \cdot x^{3}) - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

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

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

$- 1 \cdot 2 (x^{1} x^{3}) - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

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

$- 1 \cdot 2 x^{1 + 3} - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Add $1$ and $3$ .

$- 1 \cdot 2 x^{4} - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Multiply $- 1$ by $2$ .

$- 2 x^{4} - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Multiply $- 5$ by $- 1$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $- 5$ by $2$ .

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

Multiply $- 5$ by $- 5$ .

$- 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Rewrite using the commutative property of multiplication.

$- 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 7 \cdot 2 x \cdot x - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

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

Move $x$ .

$- 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 7 \cdot 2 (x \cdot x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Multiply $x$ by $x$ .

$- 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 7 \cdot 2 x^{2} - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Multiply $- 7$ by $2$ .

$- 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 14 x^{2} - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Multiply $- 5$ by $- 7$ .

$- 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 14 x^{2} + 35 x - 3 (2 x) - 3 \cdot - 5$

Multiply $2$ by $- 3$ .

$- 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 14 x^{2} + 35 x - 6 x - 3 \cdot - 5$

Multiply $- 3$ by $- 5$ .

$- 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 14 x^{2} + 35 x - 6 x + 15$

Simplify by adding terms.

Subtract $10 x^{3}$ from $5 x^{3}$ .

$- 2 x^{4} - 5 x^{3} + 25 x^{2} - 14 x^{2} + 35 x - 6 x + 15$

Subtract $14 x^{2}$ from $25 x^{2}$ .

$- 2 x^{4} - 5 x^{3} + 11 x^{2} + 35 x - 6 x + 15$

Subtract $6 x$ from $35 x$ .

$- 2 x^{4} - 5 x^{3} + 11 x^{2} + 29 x + 15$

The largest exponent is the degree of the polynomial.

$4$

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.

Simplify by multiplying through.

Apply the distributive property.

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

Simplify the expression.

Multiply $- 1$ by $3$ .

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

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

$(- x - 3) ((x + 1) (x + 1)) (2 x - 5)$

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

Apply the distributive property.

$(- x - 3) (x (x + 1) + 1 (x + 1)) (2 x - 5)$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $x$ by $1$ .

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

Multiply $1$ by $1$ .

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

Add $x$ and $x$ .

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

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

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

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $- 1$ by $1$ .

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

Multiply $2$ by $- 3$ .

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

Multiply $- 3$ by $1$ .

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

Simplify by adding terms.

Subtract $3 x^{2}$ from $- 2 x^{2}$ .

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

Subtract $6 x$ from $- x$ .

$(- x^{3} - 5 x^{2} - 7 x - 3) (2 x - 5)$

Expand $(- x^{3} - 5 x^{2} - 7 x - 3) (2 x - 5)$ by multiplying each term in the first expression by each term in the second expression.

$- x^{3} (2 x) - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$- 1 \cdot 2 x^{3} x - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

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

Move $x$ .

$- 1 \cdot 2 (x \cdot x^{3}) - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

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

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

$- 1 \cdot 2 (x^{1} x^{3}) - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

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

$- 1 \cdot 2 x^{1 + 3} - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Add $1$ and $3$ .

$- 1 \cdot 2 x^{4} - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Multiply $- 1$ by $2$ .

$- 2 x^{4} - x^{3} \cdot - 5 - 5 x^{2} (2 x) - 5 x^{2} \cdot - 5 - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Multiply $- 5$ by $- 1$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $- 5$ by $2$ .

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

Multiply $- 5$ by $- 5$ .

$- 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 7 x (2 x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Rewrite using the commutative property of multiplication.

$- 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 7 \cdot 2 x \cdot x - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

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

Move $x$ .

$- 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 7 \cdot 2 (x \cdot x) - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Multiply $x$ by $x$ .

$- 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 7 \cdot 2 x^{2} - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Multiply $- 7$ by $2$ .

$- 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 14 x^{2} - 7 x \cdot - 5 - 3 (2 x) - 3 \cdot - 5$

Multiply $- 5$ by $- 7$ .

$- 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 14 x^{2} + 35 x - 3 (2 x) - 3 \cdot - 5$

Multiply $2$ by $- 3$ .

$- 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 14 x^{2} + 35 x - 6 x - 3 \cdot - 5$

Multiply $- 3$ by $- 5$ .

$- 2 x^{4} + 5 x^{3} - 10 x^{3} + 25 x^{2} - 14 x^{2} + 35 x - 6 x + 15$

Simplify by adding terms.

Subtract $10 x^{3}$ from $5 x^{3}$ .

$- 2 x^{4} - 5 x^{3} + 25 x^{2} - 14 x^{2} + 35 x - 6 x + 15$

Subtract $14 x^{2}$ from $25 x^{2}$ .

$- 2 x^{4} - 5 x^{3} + 11 x^{2} + 35 x - 6 x + 15$

Subtract $6 x$ from $35 x$ .

$- 2 x^{4} - 5 x^{3} + 11 x^{2} + 29 x + 15$

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

$- 2 x^{4}$

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

$- 2$

Step 4

Since the leading coefficient is negative, the graph falls to the right.

Negative

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 falls to the right

Step 7

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

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