Find the End Behavior f(x)=-x(2x+1)(x+2)

$f (x) = - x (2 x + 1) (x + 2)$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Simplify terms.

Apply the distributive property.

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

Simplify the expression.

Rewrite using the commutative property of multiplication.

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

Multiply $- 1$ by $1$ .

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

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $2$ .

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

Expand $(- 2 x^{2} - x) (x + 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

$- 2 x^{2} x - 2 x^{2} \cdot 2 - x \cdot x - x \cdot 2$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $2$ by $- 2$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $2$ by $- 1$ .

$- 2 x^{3} - 4 x^{2} - x^{2} - 2 x$

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

$- 2 x^{3} - 5 x^{2} - 2 x$

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.

Simplify terms.

Apply the distributive property.

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

Simplify the expression.

Rewrite using the commutative property of multiplication.

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

Multiply $- 1$ by $1$ .

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

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $2$ .

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

Expand $(- 2 x^{2} - x) (x + 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

$- 2 x^{2} x - 2 x^{2} \cdot 2 - x \cdot x - x \cdot 2$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $2$ by $- 2$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $2$ by $- 1$ .

$- 2 x^{3} - 4 x^{2} - x^{2} - 2 x$

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

$- 2 x^{3} - 5 x^{2} - 2 x$

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

$- 2 x^{3}$

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.

Rises to the left and falls to the right

Step 7

Find the End Behavior f(x)=-x(2x+1)(x+2)

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