Algebra Examples

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

Step 1

Write $x^{4} (x^{2} - 1)$ as a function.

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

Step 2

Simplify.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 2.2.1

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

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

Step 2.2.2

Add $4$ and $2$ .

$f (x) = x^{6} + x^{4} \cdot - 1$

Step 2.3

Move $- 1$ to the left of $x^{4}$ .

$f (x) = x^{6} - 1 \cdot x^{4}$

Step 2.4

Rewrite $- 1 x^{4}$ as $- x^{4}$ .

$f (x) = x^{6} - x^{4}$

Step 3

Find $f (- x)$ .

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Step 3.1

Find $f (- x)$ by substituting $- x$ for all occurrence of $x$ in $f (x)$ .

$f (- x) = {(- x)}^{6} - {(- x)}^{4}$

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{6} x^{6} - {(- x)}^{4}$

Step 3.2.2

Raise $- 1$ to the power of $6$ .

$f (- x) = 1 x^{6} - {(- x)}^{4}$

Step 3.2.3

Multiply $x^{6}$ by $1$ .

$f (- x) = x^{6} - {(- x)}^{4}$

Step 3.2.4

Apply the product rule to $- x$ .

$f (- x) = x^{6} - ({(- 1)}^{4} x^{4})$

Step 3.2.5

Multiply $- 1$ by ${(- 1)}^{4}$ by adding the exponents.

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Step 3.2.5.1

Move ${(- 1)}^{4}$ .

$f (- x) = x^{6} + {(- 1)}^{4} \cdot (- 1 x^{4})$

Step 3.2.5.2

Multiply ${(- 1)}^{4}$ by $- 1$ .

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Step 3.2.5.2.1

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

$f (- x) = x^{6} + {(- 1)}^{4} \cdot ((- 1) x^{4})$

Step 3.2.5.2.2

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

$f (- x) = x^{6} + {(- 1)}^{4 + 1} x^{4}$

Step 3.2.5.3

Add $4$ and $1$ .

$f (- x) = x^{6} + {(- 1)}^{5} x^{4}$

Step 3.2.6

Raise $- 1$ to the power of $5$ .

$f (- x) = x^{6} - x^{4}$

Step 4

A function is even if $f (- x) = f (x)$ .

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Step 4.1

Check if $f (- x) = f (x)$ .

Step 4.2

Since $x^{6} - x^{4} = x^{6} - x^{4}$ , the function is even.

The function is even

Step 5