Find the Behavior (Leading Coefficient Test) -4(x-4)^2(x^2-1)

$- 4 {(x - 4)}^{2} (x^{2} - 1)$

Step 1

Write $- 4 {(x - 4)}^{2} (x^{2} - 1)$ as a function.

$f (x) = - 4 {(x - 4)}^{2} (x^{2} - 1)$

Step 2

Identify the degree of the function.

Simplify and reorder the polynomial.

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$- 4 ((x - 4) (x - 4)) (x^{2} - 1)$

Expand $(x - 4) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$- 4 (x (x - 4) - 4 (x - 4)) (x^{2} - 1)$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 4$ to the left of $x$ .

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

Multiply $- 4$ by $- 4$ .

$- 4 (x^{2} - 4 x - 4 x + 16) (x^{2} - 1)$

Subtract $4 x$ from $- 4 x$ .

$- 4 (x^{2} - 8 x + 16) (x^{2} - 1)$

Apply the distributive property.

$(- 4 x^{2} - 4 (- 8 x) - 4 \cdot 16) (x^{2} - 1)$

Simplify.

Multiply $- 8$ by $- 4$ .

$(- 4 x^{2} + 32 x - 4 \cdot 16) (x^{2} - 1)$

Multiply $- 4$ by $16$ .

$(- 4 x^{2} + 32 x - 64) (x^{2} - 1)$

Expand $(- 4 x^{2} + 32 x - 64) (x^{2} - 1)$ by multiplying each term in the first expression by each term in the second expression.

$- 4 x^{2} x^{2} - 4 x^{2} \cdot - 1 + 32 x \cdot x^{2} + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$- 4 (x^{2} x^{2}) - 4 x^{2} \cdot - 1 + 32 x \cdot x^{2} + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

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

$- 4 x^{2 + 2} - 4 x^{2} \cdot - 1 + 32 x \cdot x^{2} + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

Add $2$ and $2$ .

$- 4 x^{4} - 4 x^{2} \cdot - 1 + 32 x \cdot x^{2} + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

Multiply $- 1$ by $- 4$ .

$- 4 x^{4} + 4 x^{2} + 32 x \cdot x^{2} + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

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

Move $x^{2}$ .

$- 4 x^{4} + 4 x^{2} + 32 (x^{2} x) + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

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

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

$- 4 x^{4} + 4 x^{2} + 32 (x^{2} x^{1}) + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

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

$- 4 x^{4} + 4 x^{2} + 32 x^{2 + 1} + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

Add $2$ and $1$ .

$- 4 x^{4} + 4 x^{2} + 32 x^{3} + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

Multiply $- 1$ by $32$ .

$- 4 x^{4} + 4 x^{2} + 32 x^{3} - 32 x - 64 x^{2} - 64 \cdot - 1$

Multiply $- 64$ by $- 1$ .

$- 4 x^{4} + 4 x^{2} + 32 x^{3} - 32 x - 64 x^{2} + 64$

Subtract $64 x^{2}$ from $4 x^{2}$ .

$- 4 x^{4} - 60 x^{2} + 32 x^{3} - 32 x + 64$

The largest exponent is the degree of the polynomial.

$4$

Step 3

Since the degree is even, the ends of the function will point in the same direction.

Even

Step 4

Identify the leading coefficient.

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

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$- 4 ((x - 4) (x - 4)) (x^{2} - 1)$

Expand $(x - 4) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$- 4 (x (x - 4) - 4 (x - 4)) (x^{2} - 1)$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 4$ to the left of $x$ .

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

Multiply $- 4$ by $- 4$ .

$- 4 (x^{2} - 4 x - 4 x + 16) (x^{2} - 1)$

Subtract $4 x$ from $- 4 x$ .

$- 4 (x^{2} - 8 x + 16) (x^{2} - 1)$

Apply the distributive property.

$(- 4 x^{2} - 4 (- 8 x) - 4 \cdot 16) (x^{2} - 1)$

Simplify.

Multiply $- 8$ by $- 4$ .

$(- 4 x^{2} + 32 x - 4 \cdot 16) (x^{2} - 1)$

Multiply $- 4$ by $16$ .

$(- 4 x^{2} + 32 x - 64) (x^{2} - 1)$

Expand $(- 4 x^{2} + 32 x - 64) (x^{2} - 1)$ by multiplying each term in the first expression by each term in the second expression.

$- 4 x^{2} x^{2} - 4 x^{2} \cdot - 1 + 32 x \cdot x^{2} + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$- 4 (x^{2} x^{2}) - 4 x^{2} \cdot - 1 + 32 x \cdot x^{2} + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

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

$- 4 x^{2 + 2} - 4 x^{2} \cdot - 1 + 32 x \cdot x^{2} + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

Add $2$ and $2$ .

$- 4 x^{4} - 4 x^{2} \cdot - 1 + 32 x \cdot x^{2} + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

Multiply $- 1$ by $- 4$ .

$- 4 x^{4} + 4 x^{2} + 32 x \cdot x^{2} + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

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

Move $x^{2}$ .

$- 4 x^{4} + 4 x^{2} + 32 (x^{2} x) + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

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

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

$- 4 x^{4} + 4 x^{2} + 32 (x^{2} x^{1}) + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

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

$- 4 x^{4} + 4 x^{2} + 32 x^{2 + 1} + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

Add $2$ and $1$ .

$- 4 x^{4} + 4 x^{2} + 32 x^{3} + 32 x \cdot - 1 - 64 x^{2} - 64 \cdot - 1$

Multiply $- 1$ by $32$ .

$- 4 x^{4} + 4 x^{2} + 32 x^{3} - 32 x - 64 x^{2} - 64 \cdot - 1$

Multiply $- 64$ by $- 1$ .

$- 4 x^{4} + 4 x^{2} + 32 x^{3} - 32 x - 64 x^{2} + 64$

Simplify by adding terms.

Subtract $64 x^{2}$ from $4 x^{2}$ .

$- 4 x^{4} - 60 x^{2} + 32 x^{3} - 32 x + 64$

Move $- 60 x^{2}$ .

$- 4 x^{4} + 32 x^{3} - 60 x^{2} - 32 x + 64$

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

$- 4 x^{4}$

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

$- 4$

Step 5

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

Negative

Step 6

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 7

Determine the behavior.

Falls to the left and falls to the right

Step 8

Find the Behavior (Leading Coefficient Test) -4(x-4)^2(x^2-1)

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