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

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

Step 1

Write $- 2 x {(x - 1)}^{2} (x + 3)$ as a function.

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

Step 2

Identify the degree of the function.

Simplify and reorder the polynomial.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 1$ to the left of $x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Multiply $- 1$ by $- 1$ .

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

Subtract $x$ from $- x$ .

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

Apply the distributive property.

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

Simplify.

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

Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $- 2$ by $1$ .

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

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 2$ .

$(- 2 x^{3} + 4 x^{2} - 2 x) (x + 3)$

Expand $(- 2 x^{3} + 4 x^{2} - 2 x) (x + 3)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify terms.

Simplify each term.

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $3$ .

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

Multiply $3$ by $- 2$ .

$- 2 x^{4} - 6 x^{3} + 4 x^{2} x + 4 x^{2} \cdot 3 - 2 x \cdot x - 2 x \cdot 3$

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

Move $x$ .

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

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

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

$- 2 x^{4} - 6 x^{3} + 4 (x^{1} x^{2}) + 4 x^{2} \cdot 3 - 2 x \cdot x - 2 x \cdot 3$

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

$- 2 x^{4} - 6 x^{3} + 4 x^{1 + 2} + 4 x^{2} \cdot 3 - 2 x \cdot x - 2 x \cdot 3$

Add $1$ and $2$ .

$- 2 x^{4} - 6 x^{3} + 4 x^{3} + 4 x^{2} \cdot 3 - 2 x \cdot x - 2 x \cdot 3$

Multiply $3$ by $4$ .

$- 2 x^{4} - 6 x^{3} + 4 x^{3} + 12 x^{2} - 2 x \cdot x - 2 x \cdot 3$

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

Move $x$ .

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

Multiply $x$ by $x$ .

$- 2 x^{4} - 6 x^{3} + 4 x^{3} + 12 x^{2} - 2 x^{2} - 2 x \cdot 3$

Multiply $3$ by $- 2$ .

$- 2 x^{4} - 6 x^{3} + 4 x^{3} + 12 x^{2} - 2 x^{2} - 6 x$

Simplify by adding terms.

Add $- 6 x^{3}$ and $4 x^{3}$ .

$- 2 x^{4} - 2 x^{3} + 12 x^{2} - 2 x^{2} - 6 x$

Subtract $2 x^{2}$ from $12 x^{2}$ .

$- 2 x^{4} - 2 x^{3} + 10 x^{2} - 6 x$

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 ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 1$ to the left of $x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Multiply $- 1$ by $- 1$ .

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

Subtract $x$ from $- x$ .

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

Apply the distributive property.

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

Simplify.

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

Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $- 2$ by $1$ .

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

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 2$ .

$(- 2 x^{3} + 4 x^{2} - 2 x) (x + 3)$

Expand $(- 2 x^{3} + 4 x^{2} - 2 x) (x + 3)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify terms.

Simplify each term.

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $3$ .

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

Multiply $3$ by $- 2$ .

$- 2 x^{4} - 6 x^{3} + 4 x^{2} x + 4 x^{2} \cdot 3 - 2 x \cdot x - 2 x \cdot 3$

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

Move $x$ .

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

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

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

$- 2 x^{4} - 6 x^{3} + 4 (x^{1} x^{2}) + 4 x^{2} \cdot 3 - 2 x \cdot x - 2 x \cdot 3$

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

$- 2 x^{4} - 6 x^{3} + 4 x^{1 + 2} + 4 x^{2} \cdot 3 - 2 x \cdot x - 2 x \cdot 3$

Add $1$ and $2$ .

$- 2 x^{4} - 6 x^{3} + 4 x^{3} + 4 x^{2} \cdot 3 - 2 x \cdot x - 2 x \cdot 3$

Multiply $3$ by $4$ .

$- 2 x^{4} - 6 x^{3} + 4 x^{3} + 12 x^{2} - 2 x \cdot x - 2 x \cdot 3$

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

Move $x$ .

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

Multiply $x$ by $x$ .

$- 2 x^{4} - 6 x^{3} + 4 x^{3} + 12 x^{2} - 2 x^{2} - 2 x \cdot 3$

Multiply $3$ by $- 2$ .

$- 2 x^{4} - 6 x^{3} + 4 x^{3} + 12 x^{2} - 2 x^{2} - 6 x$

Simplify by adding terms.

Add $- 6 x^{3}$ and $4 x^{3}$ .

$- 2 x^{4} - 2 x^{3} + 12 x^{2} - 2 x^{2} - 6 x$

Subtract $2 x^{2}$ from $12 x^{2}$ .

$- 2 x^{4} - 2 x^{3} + 10 x^{2} - 6 x$

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 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) -2x(x-1)^2(x+3)

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