Find the End Behavior f(x)=(x+8)^2(2-x)

$f (x) = {(x + 8)}^{2} (2 - x)$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Rewrite ${(x + 8)}^{2}$ as $(x + 8) (x + 8)$ .

$(x + 8) (x + 8) (2 - x)$

Expand $(x + 8) (x + 8)$ using the FOIL Method.

Apply the distributive property.

$(x (x + 8) + 8 (x + 8)) (2 - x)$

Apply the distributive property.

$(x \cdot x + x \cdot 8 + 8 (x + 8)) (2 - x)$

Apply the distributive property.

$(x \cdot x + x \cdot 8 + 8 x + 8 \cdot 8) (2 - x)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$(x^{2} + x \cdot 8 + 8 x + 8 \cdot 8) (2 - x)$

Move $8$ to the left of $x$ .

$(x^{2} + 8 \cdot x + 8 x + 8 \cdot 8) (2 - x)$

Multiply $8$ by $8$ .

$(x^{2} + 8 x + 8 x + 64) (2 - x)$

Add $8 x$ and $8 x$ .

$(x^{2} + 16 x + 64) (2 - x)$

Expand $(x^{2} + 16 x + 64) (2 - x)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} \cdot 2 + x^{2} (- x) + 16 x \cdot 2 + 16 x (- x) + 64 \cdot 2 + 64 (- x)$

Simplify terms.

Simplify each term.

Move $2$ to the left of $x^{2}$ .

$2 \cdot x^{2} + x^{2} (- x) + 16 x \cdot 2 + 16 x (- x) + 64 \cdot 2 + 64 (- x)$

Rewrite using the commutative property of multiplication.

$2 x^{2} - x^{2} x + 16 x \cdot 2 + 16 x (- x) + 64 \cdot 2 + 64 (- x)$

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

Move $x$ .

$2 x^{2} - (x \cdot x^{2}) + 16 x \cdot 2 + 16 x (- x) + 64 \cdot 2 + 64 (- x)$

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

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

$2 x^{2} - (x^{1} x^{2}) + 16 x \cdot 2 + 16 x (- x) + 64 \cdot 2 + 64 (- x)$

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

$2 x^{2} - x^{1 + 2} + 16 x \cdot 2 + 16 x (- x) + 64 \cdot 2 + 64 (- x)$

Add $1$ and $2$ .

$2 x^{2} - x^{3} + 16 x \cdot 2 + 16 x (- x) + 64 \cdot 2 + 64 (- x)$

Multiply $2$ by $16$ .

$2 x^{2} - x^{3} + 32 x + 16 x (- x) + 64 \cdot 2 + 64 (- x)$

Rewrite using the commutative property of multiplication.

$2 x^{2} - x^{3} + 32 x + 16 \cdot - 1 x \cdot x + 64 \cdot 2 + 64 (- x)$

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

Move $x$ .

$2 x^{2} - x^{3} + 32 x + 16 \cdot - 1 (x \cdot x) + 64 \cdot 2 + 64 (- x)$

Multiply $x$ by $x$ .

$2 x^{2} - x^{3} + 32 x + 16 \cdot - 1 x^{2} + 64 \cdot 2 + 64 (- x)$

Multiply $16$ by $- 1$ .

$2 x^{2} - x^{3} + 32 x - 16 x^{2} + 64 \cdot 2 + 64 (- x)$

Multiply $64$ by $2$ .

$2 x^{2} - x^{3} + 32 x - 16 x^{2} + 128 + 64 (- x)$

Multiply $- 1$ by $64$ .

$2 x^{2} - x^{3} + 32 x - 16 x^{2} + 128 - 64 x$

Simplify by adding terms.

Subtract $16 x^{2}$ from $2 x^{2}$ .

$- 14 x^{2} - x^{3} + 32 x + 128 - 64 x$

Subtract $64 x$ from $32 x$ .

$- 14 x^{2} - x^{3} - 32 x + 128$

The largest exponent is the degree of the polynomial.

$3$

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 ${(x + 8)}^{2}$ as $(x + 8) (x + 8)$ .

$(x + 8) (x + 8) (2 - x)$

Expand $(x + 8) (x + 8)$ using the FOIL Method.

Apply the distributive property.

$(x (x + 8) + 8 (x + 8)) (2 - x)$

Apply the distributive property.

$(x \cdot x + x \cdot 8 + 8 (x + 8)) (2 - x)$

Apply the distributive property.

$(x \cdot x + x \cdot 8 + 8 x + 8 \cdot 8) (2 - x)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$(x^{2} + x \cdot 8 + 8 x + 8 \cdot 8) (2 - x)$

Move $8$ to the left of $x$ .

$(x^{2} + 8 \cdot x + 8 x + 8 \cdot 8) (2 - x)$

Multiply $8$ by $8$ .

$(x^{2} + 8 x + 8 x + 64) (2 - x)$

Add $8 x$ and $8 x$ .

$(x^{2} + 16 x + 64) (2 - x)$

Expand $(x^{2} + 16 x + 64) (2 - x)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} \cdot 2 + x^{2} (- x) + 16 x \cdot 2 + 16 x (- x) + 64 \cdot 2 + 64 (- x)$

Simplify terms.

Simplify each term.

Move $2$ to the left of $x^{2}$ .

$2 \cdot x^{2} + x^{2} (- x) + 16 x \cdot 2 + 16 x (- x) + 64 \cdot 2 + 64 (- x)$

Rewrite using the commutative property of multiplication.

$2 x^{2} - x^{2} x + 16 x \cdot 2 + 16 x (- x) + 64 \cdot 2 + 64 (- x)$

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

Move $x$ .

$2 x^{2} - (x \cdot x^{2}) + 16 x \cdot 2 + 16 x (- x) + 64 \cdot 2 + 64 (- x)$

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

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

$2 x^{2} - (x^{1} x^{2}) + 16 x \cdot 2 + 16 x (- x) + 64 \cdot 2 + 64 (- x)$

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

$2 x^{2} - x^{1 + 2} + 16 x \cdot 2 + 16 x (- x) + 64 \cdot 2 + 64 (- x)$

Add $1$ and $2$ .

$2 x^{2} - x^{3} + 16 x \cdot 2 + 16 x (- x) + 64 \cdot 2 + 64 (- x)$

Multiply $2$ by $16$ .

$2 x^{2} - x^{3} + 32 x + 16 x (- x) + 64 \cdot 2 + 64 (- x)$

Rewrite using the commutative property of multiplication.

$2 x^{2} - x^{3} + 32 x + 16 \cdot - 1 x \cdot x + 64 \cdot 2 + 64 (- x)$

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

Move $x$ .

$2 x^{2} - x^{3} + 32 x + 16 \cdot - 1 (x \cdot x) + 64 \cdot 2 + 64 (- x)$

Multiply $x$ by $x$ .

$2 x^{2} - x^{3} + 32 x + 16 \cdot - 1 x^{2} + 64 \cdot 2 + 64 (- x)$

Multiply $16$ by $- 1$ .

$2 x^{2} - x^{3} + 32 x - 16 x^{2} + 64 \cdot 2 + 64 (- x)$

Multiply $64$ by $2$ .

$2 x^{2} - x^{3} + 32 x - 16 x^{2} + 128 + 64 (- x)$

Multiply $- 1$ by $64$ .

$2 x^{2} - x^{3} + 32 x - 16 x^{2} + 128 - 64 x$

Simplify by adding terms.

Subtract $16 x^{2}$ from $2 x^{2}$ .

$- 14 x^{2} - x^{3} + 32 x + 128 - 64 x$

Subtract $64 x$ from $32 x$ .

$- 14 x^{2} - x^{3} - 32 x + 128$

Reorder $- 14 x^{2}$ and $- x^{3}$ .

$- x^{3} - 14 x^{2} - 32 x + 128$

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

$- x^{3}$

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

$- 1$

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.

Rises to the left and falls to the right

Step 7

Find the End Behavior f(x)=(x+8)^2(2-x)

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