Precalculus Examples

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

Step 1

Identify the degree of the function.

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Step 1.1

Simplify and reorder the polynomial.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

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

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Step 1.1.2.1

Move $x$ .

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

Step 1.1.2.2

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

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Step 1.1.2.2.1

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

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

Step 1.1.2.2.2

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

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

Step 1.1.2.3

Add $1$ and $2$ .

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

Step 1.1.3

Multiply $4$ by $- 1$ .

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

Step 1.1.4

Expand $(- x^{3} - 4 x^{2}) (x^{2} - 1)$ using the FOIL Method.

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Step 1.1.4.1

Apply the distributive property.

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

Step 1.1.4.2

Apply the distributive property.

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

Step 1.1.4.3

Apply the distributive property.

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

Step 1.1.5

Simplify each term.

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Step 1.1.5.1

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

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Step 1.1.5.1.1

Move $x^{2}$ .

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

Step 1.1.5.1.2

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

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

Step 1.1.5.1.3

Add $2$ and $3$ .

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

Step 1.1.5.2

Multiply $- x^{3} \cdot - 1$ .

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Step 1.1.5.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.1.5.2.2

Multiply $x^{3}$ by $1$ .

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

Step 1.1.5.3

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

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Step 1.1.5.3.1

Move $x^{2}$ .

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

Step 1.1.5.3.2

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

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

Step 1.1.5.3.3

Add $2$ and $2$ .

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

Step 1.1.5.4

Multiply $- 1$ by $- 4$ .

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

Step 1.2

The largest exponent is the degree of the polynomial.

$5$

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.

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Step 3.1

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

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Step 3.1.1

Apply the distributive property.

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

Step 3.1.2

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

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Step 3.1.2.1

Move $x$ .

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

Step 3.1.2.2

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

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Step 3.1.2.2.1

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

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

Step 3.1.2.2.2

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

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

Step 3.1.2.3

Add $1$ and $2$ .

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

Step 3.1.3

Multiply $4$ by $- 1$ .

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

Step 3.1.4

Expand $(- x^{3} - 4 x^{2}) (x^{2} - 1)$ using the FOIL Method.

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Step 3.1.4.1

Apply the distributive property.

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

Step 3.1.4.2

Apply the distributive property.

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

Step 3.1.4.3

Apply the distributive property.

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

Step 3.1.5

Simplify terms.

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Step 3.1.5.1

Simplify each term.

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Step 3.1.5.1.1

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

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Step 3.1.5.1.1.1

Move $x^{2}$ .

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

Step 3.1.5.1.1.2

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

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

Step 3.1.5.1.1.3

Add $2$ and $3$ .

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

Step 3.1.5.1.2

Multiply $- x^{3} \cdot - 1$ .

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Step 3.1.5.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.5.1.2.2

Multiply $x^{3}$ by $1$ .

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

Step 3.1.5.1.3

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

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Step 3.1.5.1.3.1

Move $x^{2}$ .

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

Step 3.1.5.1.3.2

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

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

Step 3.1.5.1.3.3

Add $2$ and $2$ .

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

Step 3.1.5.1.4

Multiply $- 1$ by $- 4$ .

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

Step 3.1.5.2

Move $x^{3}$ .

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

Step 3.2

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

$- x^{5}$

Step 3.3

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.

Rises to the left and falls to the right

Step 7