Find the End Behavior f(x)=-x^2(x-2)(x+5)

$f (x) = - x^{2} (x - 2) (x + 5)$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Apply the distributive property.

$(- x^{2} x - x^{2} \cdot - 2) (x + 5)$

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

Move $x$ .

$(- (x \cdot x^{2}) - x^{2} \cdot - 2) (x + 5)$

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

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

$(- (x^{1} x^{2}) - x^{2} \cdot - 2) (x + 5)$

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

$(- x^{1 + 2} - x^{2} \cdot - 2) (x + 5)$

Add $1$ and $2$ .

$(- x^{3} - x^{2} \cdot - 2) (x + 5)$

Multiply $- 2$ by $- 1$ .

$(- x^{3} + 2 x^{2}) (x + 5)$

Expand $(- x^{3} + 2 x^{2}) (x + 5)$ using the FOIL Method.

Apply the distributive property.

$- x^{3} (x + 5) + 2 x^{2} (x + 5)$

Apply the distributive property.

$- x^{3} x - x^{3} \cdot 5 + 2 x^{2} (x + 5)$

Apply the distributive property.

$- x^{3} x - x^{3} \cdot 5 + 2 x^{2} x + 2 x^{2} \cdot 5$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$- (x \cdot x^{3}) - x^{3} \cdot 5 + 2 x^{2} x + 2 x^{2} \cdot 5$

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

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

$- (x^{1} x^{3}) - x^{3} \cdot 5 + 2 x^{2} x + 2 x^{2} \cdot 5$

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

$- x^{1 + 3} - x^{3} \cdot 5 + 2 x^{2} x + 2 x^{2} \cdot 5$

Add $1$ and $3$ .

$- x^{4} - x^{3} \cdot 5 + 2 x^{2} x + 2 x^{2} \cdot 5$

Multiply $5$ by $- 1$ .

$- x^{4} - 5 x^{3} + 2 x^{2} x + 2 x^{2} \cdot 5$

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

Move $x$ .

$- x^{4} - 5 x^{3} + 2 (x \cdot x^{2}) + 2 x^{2} \cdot 5$

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

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

$- x^{4} - 5 x^{3} + 2 (x^{1} x^{2}) + 2 x^{2} \cdot 5$

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

$- x^{4} - 5 x^{3} + 2 x^{1 + 2} + 2 x^{2} \cdot 5$

Add $1$ and $2$ .

$- x^{4} - 5 x^{3} + 2 x^{3} + 2 x^{2} \cdot 5$

Multiply $5$ by $2$ .

$- x^{4} - 5 x^{3} + 2 x^{3} + 10 x^{2}$

Add $- 5 x^{3}$ and $2 x^{3}$ .

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

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.

Apply the distributive property.

$(- x^{2} x - x^{2} \cdot - 2) (x + 5)$

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

Move $x$ .

$(- (x \cdot x^{2}) - x^{2} \cdot - 2) (x + 5)$

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

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

$(- (x^{1} x^{2}) - x^{2} \cdot - 2) (x + 5)$

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

$(- x^{1 + 2} - x^{2} \cdot - 2) (x + 5)$

Add $1$ and $2$ .

$(- x^{3} - x^{2} \cdot - 2) (x + 5)$

Multiply $- 2$ by $- 1$ .

$(- x^{3} + 2 x^{2}) (x + 5)$

Expand $(- x^{3} + 2 x^{2}) (x + 5)$ using the FOIL Method.

Apply the distributive property.

$- x^{3} (x + 5) + 2 x^{2} (x + 5)$

Apply the distributive property.

$- x^{3} x - x^{3} \cdot 5 + 2 x^{2} (x + 5)$

Apply the distributive property.

$- x^{3} x - x^{3} \cdot 5 + 2 x^{2} x + 2 x^{2} \cdot 5$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$- (x \cdot x^{3}) - x^{3} \cdot 5 + 2 x^{2} x + 2 x^{2} \cdot 5$

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

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

$- (x^{1} x^{3}) - x^{3} \cdot 5 + 2 x^{2} x + 2 x^{2} \cdot 5$

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

$- x^{1 + 3} - x^{3} \cdot 5 + 2 x^{2} x + 2 x^{2} \cdot 5$

Add $1$ and $3$ .

$- x^{4} - x^{3} \cdot 5 + 2 x^{2} x + 2 x^{2} \cdot 5$

Multiply $5$ by $- 1$ .

$- x^{4} - 5 x^{3} + 2 x^{2} x + 2 x^{2} \cdot 5$

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

Move $x$ .

$- x^{4} - 5 x^{3} + 2 (x \cdot x^{2}) + 2 x^{2} \cdot 5$

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

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

$- x^{4} - 5 x^{3} + 2 (x^{1} x^{2}) + 2 x^{2} \cdot 5$

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

$- x^{4} - 5 x^{3} + 2 x^{1 + 2} + 2 x^{2} \cdot 5$

Add $1$ and $2$ .

$- x^{4} - 5 x^{3} + 2 x^{3} + 2 x^{2} \cdot 5$

Multiply $5$ by $2$ .

$- x^{4} - 5 x^{3} + 2 x^{3} + 10 x^{2}$

Add $- 5 x^{3}$ and $2 x^{3}$ .

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

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

$- x^{4}$

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.

Falls to the left and falls to the right

Step 7

Find the End Behavior f(x)=-x^2(x-2)(x+5)

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