Algebra Examples

$x^{12} + x^{9}$

Step 1

Write $x^{12} + x^{9}$ as a function.

$f (x) = x^{12} + x^{9}$

Step 2

Find $f (- x)$ .

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Step 2.1

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

$f (- x) = {(- x)}^{12} + {(- x)}^{9}$

Step 2.2

Simplify each term.

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Step 2.2.1

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{12} x^{12} + {(- x)}^{9}$

Step 2.2.2

Raise $- 1$ to the power of $12$ .

$f (- x) = 1 x^{12} + {(- x)}^{9}$

Step 2.2.3

Multiply $x^{12}$ by $1$ .

$f (- x) = x^{12} + {(- x)}^{9}$

Step 2.2.4

Apply the product rule to $- x$ .

$f (- x) = x^{12} + {(- 1)}^{9} x^{9}$

Step 2.2.5

Raise $- 1$ to the power of $9$ .

$f (- x) = x^{12} - x^{9}$

Step 3

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

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Step 3.1

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

Step 3.2

Since $x^{12} - x^{9}$ $\neq$ $x^{12} + x^{9}$ , the function is not even.

The function is not even

Step 4

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

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Step 4.1

Find $- f (x)$ .

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Step 4.1.1

Multiply $x^{12} + x^{9}$ by $- 1$ .

$- f (x) = - (x^{12} + x^{9})$

Step 4.1.2

Apply the distributive property.

$- f (x) = - x^{12} - x^{9}$

Step 4.2

Since $x^{12} - x^{9}$ $\neq$ $- x^{12} - x^{9}$ , the function is not odd.

The function is not odd

Step 5

The function is neither odd nor even

Step 6