Finite Math Examples

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

Step 1

Simplify $f (x)$ .

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Step 1.1

Apply the distributive property.

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

Step 1.2

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

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Step 1.2.1

Move $x$ .

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

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

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

Step 1.2.3

Add $1$ and $2$ .

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

Step 1.3

Multiply $- x^{2} \cdot - 1$ .

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Step 1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.3.2

Multiply $x^{2}$ by $1$ .

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

Step 1.4

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

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Step 1.4.1

Apply the distributive property.

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

Step 1.4.2

Apply the distributive property.

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

Step 1.4.3

Apply the distributive property.

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

Step 1.5

Simplify and combine like terms.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

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

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Step 1.5.1.1.1

Move $x$ .

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

Step 1.5.1.1.2

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

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Step 1.5.1.1.2.1

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

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

Step 1.5.1.1.2.2

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

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

Step 1.5.1.1.3

Add $1$ and $3$ .

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

Step 1.5.1.2

Multiply $5$ by $- 1$ .

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

Step 1.5.1.3

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

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Step 1.5.1.3.1

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

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Step 1.5.1.3.1.1

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

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

Step 1.5.1.3.1.2

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

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

Step 1.5.1.3.2

Add $2$ and $1$ .

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

Step 1.5.1.4

Move $5$ to the left of $x^{2}$ .

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

Step 1.5.2

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

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

Step 2

The word linear is used for a straight line. A linear function is a function of a straight line, which means that the degree of a linear function must be $0$ or $1$ . In this case, The degree of $f (x) = - x^{4} - 4 x^{3} + 5 x^{2}$ is $4$ , which makes the function a nonlinear function.

$f (x) = - x^{4} - 4 x^{3} + 5 x^{2}$ is not a linear function