Algebra Examples

$f (x) = - 2 x^{3} (x - 1) (x + 5)$

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.

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

Step 1.1.2

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

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Step 1.1.2.1

Move $x$ .

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

Step 1.1.2.2

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

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Step 1.1.2.2.1

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

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

Step 1.1.2.2.2

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

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

Step 1.1.2.3

Add $1$ and $3$ .

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

Step 1.1.3

Multiply $- 1$ by $- 2$ .

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

Step 1.1.4

Expand $(- 2 x^{4} + 2 x^{3}) (x + 5)$ using the FOIL Method.

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Step 1.1.4.1

Apply the distributive property.

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

Step 1.1.4.2

Apply the distributive property.

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

Step 1.1.4.3

Apply the distributive property.

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

Step 1.1.5

Simplify and combine like terms.

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Step 1.1.5.1

Simplify each term.

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Step 1.1.5.1.1

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 1.1.5.1.1.1

Move $x$ .

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

Step 1.1.5.1.1.2

Multiply $x$ by $x^{4}$ .

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Step 1.1.5.1.1.2.1

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

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

Step 1.1.5.1.1.2.2

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

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

Step 1.1.5.1.1.3

Add $1$ and $4$ .

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

Step 1.1.5.1.2

Multiply $5$ by $- 2$ .

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

Step 1.1.5.1.3

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

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Step 1.1.5.1.3.1

Move $x$ .

$- 2 x^{5} - 10 x^{4} + 2 (x \cdot x^{3}) + 2 x^{3} \cdot 5$

Step 1.1.5.1.3.2

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

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Step 1.1.5.1.3.2.1

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

$- 2 x^{5} - 10 x^{4} + 2 (x^{1} x^{3}) + 2 x^{3} \cdot 5$

Step 1.1.5.1.3.2.2

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

$- 2 x^{5} - 10 x^{4} + 2 x^{1 + 3} + 2 x^{3} \cdot 5$

Step 1.1.5.1.3.3

Add $1$ and $3$ .

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

Step 1.1.5.1.4

Multiply $5$ by $2$ .

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

Step 1.1.5.2

Add $- 10 x^{4}$ and $2 x^{4}$ .

$- 2 x^{5} - 8 x^{4} + 10 x^{3}$

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.

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

Step 3.1.2

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

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Step 3.1.2.1

Move $x$ .

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

Step 3.1.2.2

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

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Step 3.1.2.2.1

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

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

Step 3.1.2.2.2

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

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

Step 3.1.2.3

Add $1$ and $3$ .

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

Step 3.1.3

Multiply $- 1$ by $- 2$ .

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

Step 3.1.4

Expand $(- 2 x^{4} + 2 x^{3}) (x + 5)$ using the FOIL Method.

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Step 3.1.4.1

Apply the distributive property.

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

Step 3.1.4.2

Apply the distributive property.

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

Step 3.1.4.3

Apply the distributive property.

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

Step 3.1.5

Simplify and combine like terms.

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Step 3.1.5.1

Simplify each term.

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Step 3.1.5.1.1

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 3.1.5.1.1.1

Move $x$ .

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

Step 3.1.5.1.1.2

Multiply $x$ by $x^{4}$ .

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Step 3.1.5.1.1.2.1

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

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

Step 3.1.5.1.1.2.2

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

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

Step 3.1.5.1.1.3

Add $1$ and $4$ .

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

Step 3.1.5.1.2

Multiply $5$ by $- 2$ .

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

Step 3.1.5.1.3

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

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Step 3.1.5.1.3.1

Move $x$ .

$- 2 x^{5} - 10 x^{4} + 2 (x \cdot x^{3}) + 2 x^{3} \cdot 5$

Step 3.1.5.1.3.2

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

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Step 3.1.5.1.3.2.1

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

$- 2 x^{5} - 10 x^{4} + 2 (x^{1} x^{3}) + 2 x^{3} \cdot 5$

Step 3.1.5.1.3.2.2

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

$- 2 x^{5} - 10 x^{4} + 2 x^{1 + 3} + 2 x^{3} \cdot 5$

Step 3.1.5.1.3.3

Add $1$ and $3$ .

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

Step 3.1.5.1.4

Multiply $5$ by $2$ .

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

Step 3.1.5.2

Add $- 10 x^{4}$ and $2 x^{4}$ .

$- 2 x^{5} - 8 x^{4} + 10 x^{3}$

Step 3.2

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

$- 2 x^{5}$

Step 3.3

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