Find the Behavior (Leading Coefficient Test) f(x)=-2(x-1)(x+3)^3

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

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Simplify by multiplying through.

Apply the distributive property.

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

Multiply $- 2$ by $- 1$ .

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

Use the Binomial Theorem.

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

Simplify each term.

Multiply $3$ by $3$ .

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

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

Move $3^{2}$ .

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

Multiply $3^{2}$ by $3$ .

Raise $3$ to the power of $1$ .

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

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

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

Add $2$ and $1$ .

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

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

$(- 2 x + 2) (x^{3} + 9 x^{2} + 27 x + 3^{3})$

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

$(- 2 x + 2) (x^{3} + 9 x^{2} + 27 x + 27)$

Expand $(- 2 x + 2) (x^{3} + 9 x^{2} + 27 x + 27)$ by multiplying each term in the first expression by each term in the second expression.

$- 2 x \cdot x^{3} - 2 x (9 x^{2}) - 2 x (27 x) - 2 x \cdot 27 + 2 x^{3} + 2 (9 x^{2}) + 2 (27 x) + 2 \cdot 27$

Simplify terms.

Combine the opposite terms in $- 2 x \cdot x^{3} - 2 x (9 x^{2}) - 2 x (27 x) - 2 x \cdot 27 + 2 x^{3} + 2 (9 x^{2}) + 2 (27 x) + 2 \cdot 27$ .

Reorder the factors in the terms $- 2 x \cdot 27$ and $2 (27 x)$ .

$- 2 x \cdot x^{3} - 2 x (9 x^{2}) - 2 x (27 x) - 2 \cdot 27 x + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27 x + 2 \cdot 27$

Add $- 2 \cdot 27 x$ and $2 \cdot 27 x$ .

$- 2 x \cdot x^{3} - 2 x (9 x^{2}) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 0 + 2 \cdot 27$

Add $- 2 x \cdot x^{3} - 2 x (9 x^{2}) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2})$ and $0$ .

$- 2 x \cdot x^{3} - 2 x (9 x^{2}) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Simplify each term.

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

Move $x^{3}$ .

$- 2 (x^{3} x) - 2 x (9 x^{2}) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

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

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

$- 2 (x^{3} x^{1}) - 2 x (9 x^{2}) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

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

$- 2 x^{3 + 1} - 2 x (9 x^{2}) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Add $3$ and $1$ .

$- 2 x^{4} - 2 x (9 x^{2}) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Rewrite using the commutative property of multiplication.

$- 2 x^{4} - 2 \cdot 9 x \cdot x^{2} - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

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

Move $x^{2}$ .

$- 2 x^{4} - 2 \cdot 9 (x^{2} x) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

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

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

$- 2 x^{4} - 2 \cdot 9 (x^{2} x^{1}) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

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

$- 2 x^{4} - 2 \cdot 9 x^{2 + 1} - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Add $2$ and $1$ .

$- 2 x^{4} - 2 \cdot 9 x^{3} - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Multiply $- 2$ by $9$ .

$- 2 x^{4} - 18 x^{3} - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Rewrite using the commutative property of multiplication.

$- 2 x^{4} - 18 x^{3} - 2 \cdot 27 x \cdot x + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

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

Move $x$ .

$- 2 x^{4} - 18 x^{3} - 2 \cdot 27 (x \cdot x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Multiply $x$ by $x$ .

$- 2 x^{4} - 18 x^{3} - 2 \cdot 27 x^{2} + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Multiply $- 2$ by $27$ .

$- 2 x^{4} - 18 x^{3} - 54 x^{2} + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Multiply $9$ by $2$ .

$- 2 x^{4} - 18 x^{3} - 54 x^{2} + 2 x^{3} + 18 x^{2} + 2 \cdot 27$

Multiply $2$ by $27$ .

$- 2 x^{4} - 18 x^{3} - 54 x^{2} + 2 x^{3} + 18 x^{2} + 54$

Simplify by adding terms.

Add $- 18 x^{3}$ and $2 x^{3}$ .

$- 2 x^{4} - 16 x^{3} - 54 x^{2} + 18 x^{2} + 54$

Add $- 54 x^{2}$ and $18 x^{2}$ .

$- 2 x^{4} - 16 x^{3} - 36 x^{2} + 54$

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.

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

Multiply $- 2$ by $- 1$ .

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

Use the Binomial Theorem.

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

Simplify each term.

Multiply $3$ by $3$ .

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

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

Move $3^{2}$ .

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

Multiply $3^{2}$ by $3$ .

Raise $3$ to the power of $1$ .

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

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

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

Add $2$ and $1$ .

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

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

$(- 2 x + 2) (x^{3} + 9 x^{2} + 27 x + 3^{3})$

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

$(- 2 x + 2) (x^{3} + 9 x^{2} + 27 x + 27)$

Expand $(- 2 x + 2) (x^{3} + 9 x^{2} + 27 x + 27)$ by multiplying each term in the first expression by each term in the second expression.

$- 2 x \cdot x^{3} - 2 x (9 x^{2}) - 2 x (27 x) - 2 x \cdot 27 + 2 x^{3} + 2 (9 x^{2}) + 2 (27 x) + 2 \cdot 27$

Simplify terms.

Combine the opposite terms in $- 2 x \cdot x^{3} - 2 x (9 x^{2}) - 2 x (27 x) - 2 x \cdot 27 + 2 x^{3} + 2 (9 x^{2}) + 2 (27 x) + 2 \cdot 27$ .

Reorder the factors in the terms $- 2 x \cdot 27$ and $2 (27 x)$ .

$- 2 x \cdot x^{3} - 2 x (9 x^{2}) - 2 x (27 x) - 2 \cdot 27 x + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27 x + 2 \cdot 27$

Add $- 2 \cdot 27 x$ and $2 \cdot 27 x$ .

$- 2 x \cdot x^{3} - 2 x (9 x^{2}) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 0 + 2 \cdot 27$

Add $- 2 x \cdot x^{3} - 2 x (9 x^{2}) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2})$ and $0$ .

$- 2 x \cdot x^{3} - 2 x (9 x^{2}) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Simplify each term.

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

Move $x^{3}$ .

$- 2 (x^{3} x) - 2 x (9 x^{2}) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

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

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

$- 2 (x^{3} x^{1}) - 2 x (9 x^{2}) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

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

$- 2 x^{3 + 1} - 2 x (9 x^{2}) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Add $3$ and $1$ .

$- 2 x^{4} - 2 x (9 x^{2}) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Rewrite using the commutative property of multiplication.

$- 2 x^{4} - 2 \cdot 9 x \cdot x^{2} - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

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

Move $x^{2}$ .

$- 2 x^{4} - 2 \cdot 9 (x^{2} x) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

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

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

$- 2 x^{4} - 2 \cdot 9 (x^{2} x^{1}) - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

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

$- 2 x^{4} - 2 \cdot 9 x^{2 + 1} - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Add $2$ and $1$ .

$- 2 x^{4} - 2 \cdot 9 x^{3} - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Multiply $- 2$ by $9$ .

$- 2 x^{4} - 18 x^{3} - 2 x (27 x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Rewrite using the commutative property of multiplication.

$- 2 x^{4} - 18 x^{3} - 2 \cdot 27 x \cdot x + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

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

Move $x$ .

$- 2 x^{4} - 18 x^{3} - 2 \cdot 27 (x \cdot x) + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Multiply $x$ by $x$ .

$- 2 x^{4} - 18 x^{3} - 2 \cdot 27 x^{2} + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Multiply $- 2$ by $27$ .

$- 2 x^{4} - 18 x^{3} - 54 x^{2} + 2 x^{3} + 2 (9 x^{2}) + 2 \cdot 27$

Multiply $9$ by $2$ .

$- 2 x^{4} - 18 x^{3} - 54 x^{2} + 2 x^{3} + 18 x^{2} + 2 \cdot 27$

Multiply $2$ by $27$ .

$- 2 x^{4} - 18 x^{3} - 54 x^{2} + 2 x^{3} + 18 x^{2} + 54$

Simplify by adding terms.

Add $- 18 x^{3}$ and $2 x^{3}$ .

$- 2 x^{4} - 16 x^{3} - 54 x^{2} + 18 x^{2} + 54$

Add $- 54 x^{2}$ and $18 x^{2}$ .

$- 2 x^{4} - 16 x^{3} - 36 x^{2} + 54$

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 Behavior (Leading Coefficient Test) f(x)=-2(x-1)(x+3)^3

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