Algebra Examples

$- 9 x^{10} - 6 x^{6} + 5 x^{4} - x^{2}$

Step 1

Write $- 9 x^{10} - 6 x^{6} + 5 x^{4} - x^{2}$ as a function.

$f (x) = - 9 x^{10} - 6 x^{6} + 5 x^{4} - x^{2}$

Step 2

Find $f (- x)$ .

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Step 2.1

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

$f (- x) = - 9 {(- x)}^{10} - 6 {(- x)}^{6} + 5 {(- x)}^{4} - {(- x)}^{2}$

Step 2.2

Simplify each term.

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Step 2.2.1

Apply the product rule to $- x$ .

$f (- x) = - 9 ({(- 1)}^{10} x^{10}) - 6 {(- x)}^{6} + 5 {(- x)}^{4} - {(- x)}^{2}$

Step 2.2.2

Raise $- 1$ to the power of $10$ .

$f (- x) = - 9 (1 x^{10}) - 6 {(- x)}^{6} + 5 {(- x)}^{4} - {(- x)}^{2}$

Step 2.2.3

Multiply $x^{10}$ by $1$ .

$f (- x) = - 9 x^{10} - 6 {(- x)}^{6} + 5 {(- x)}^{4} - {(- x)}^{2}$

Step 2.2.4

Apply the product rule to $- x$ .

$f (- x) = - 9 x^{10} - 6 ({(- 1)}^{6} x^{6}) + 5 {(- x)}^{4} - {(- x)}^{2}$

Step 2.2.5

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

$f (- x) = - 9 x^{10} - 6 (1 x^{6}) + 5 {(- x)}^{4} - {(- x)}^{2}$

Step 2.2.6

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

$f (- x) = - 9 x^{10} - 6 x^{6} + 5 {(- x)}^{4} - {(- x)}^{2}$

Step 2.2.7

Apply the product rule to $- x$ .

$f (- x) = - 9 x^{10} - 6 x^{6} + 5 ({(- 1)}^{4} x^{4}) - {(- x)}^{2}$

Step 2.2.8

Raise $- 1$ to the power of $4$ .

$f (- x) = - 9 x^{10} - 6 x^{6} + 5 (1 x^{4}) - {(- x)}^{2}$

Step 2.2.9

Multiply $x^{4}$ by $1$ .

$f (- x) = - 9 x^{10} - 6 x^{6} + 5 x^{4} - {(- x)}^{2}$

Step 2.2.10

Apply the product rule to $- x$ .

$f (- x) = - 9 x^{10} - 6 x^{6} + 5 x^{4} - ({(- 1)}^{2} x^{2})$

Step 2.2.11

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 2.2.11.1

Move ${(- 1)}^{2}$ .

$f (- x) = - 9 x^{10} - 6 x^{6} + 5 x^{4} + {(- 1)}^{2} \cdot (- 1 x^{2})$

Step 2.2.11.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 2.2.11.2.1

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

$f (- x) = - 9 x^{10} - 6 x^{6} + 5 x^{4} + {(- 1)}^{2} \cdot ((- 1) x^{2})$

Step 2.2.11.2.2

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

$f (- x) = - 9 x^{10} - 6 x^{6} + 5 x^{4} + {(- 1)}^{2 + 1} x^{2}$

Step 2.2.11.3

Add $2$ and $1$ .

$f (- x) = - 9 x^{10} - 6 x^{6} + 5 x^{4} + {(- 1)}^{3} x^{2}$

Step 2.2.12

Raise $- 1$ to the power of $3$ .

$f (- x) = - 9 x^{10} - 6 x^{6} + 5 x^{4} - x^{2}$

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 $- 9 x^{10} - 6 x^{6} + 5 x^{4} - x^{2} = - 9 x^{10} - 6 x^{6} + 5 x^{4} - x^{2}$ , the function is even.

The function is even

Step 4