Determine if Odd, Even, or Neither 3x^4-6x^3

$3 x^{4} - 6 x^{3}$

Step 1

Write $3 x^{4} - 6 x^{3}$ as a function.

$f (x) = 3 x^{4} - 6 x^{3}$

Step 2

Find $f (- x)$ .

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

$f (- x) = 3 {(- x)}^{4} - 6 {(- x)}^{3}$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = 3 ({(- 1)}^{4} x^{4}) - 6 {(- x)}^{3}$

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

$f (- x) = 3 (1 x^{4}) - 6 {(- x)}^{3}$

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

$f (- x) = 3 x^{4} - 6 {(- x)}^{3}$

Apply the product rule to $- x$ .

$f (- x) = 3 x^{4} - 6 ({(- 1)}^{3} x^{3})$

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

$f (- x) = 3 x^{4} - 6 (- x^{3})$

Multiply $- 1$ by $- 6$ .

$f (- x) = 3 x^{4} + 6 x^{3}$

Step 3

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

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

Since $3 x^{4} + 6 x^{3}$ $\neq$ $3 x^{4} - 6 x^{3}$ , the function is not even.

The function is not even

Step 4

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

Find $- f (x)$ .

Multiply $3 x^{4} - 6 x^{3}$ by $- 1$ .

$- f (x) = - (3 x^{4} - 6 x^{3})$

Apply the distributive property.

$- f (x) = - (3 x^{4}) - (- 6 x^{3})$

Multiply.

Multiply $3$ by $- 1$ .

$- f (x) = - 3 x^{4} - (- 6 x^{3})$

Multiply $- 6$ by $- 1$ .

$- f (x) = - 3 x^{4} + 6 x^{3}$

Since $3 x^{4} + 6 x^{3}$ $\neq$ $- 3 x^{4} + 6 x^{3}$ , the function is not odd.

The function is not odd

Step 5

The function is neither odd nor even

Step 6