Determine if Odd, Even, or Neither 3x^3+2x^2+1

$3 x^{3} + 2 x^{2} + 1$

Step 1

Write $3 x^{3} + 2 x^{2} + 1$ as a function.

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

Step 2

Find $f (- x)$ .

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

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

Simplify each term.

Apply the product rule to $- x$ .

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

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

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

Multiply $- 1$ by $3$ .

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

Apply the product rule to $- x$ .

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

Raise $- 1$ to the power of $2$ .

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

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

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

Step 3

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

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

Since $- 3 x^{3} + 2 x^{2} + 1$ $\neq$ $3 x^{3} + 2 x^{2} + 1$ , 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 $3 x^{3} + 2 x^{2} + 1$ by $- 1$ .

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

Apply the distributive property.

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

Simplify.

Multiply $3$ by $- 1$ .

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

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $1$ .

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

Since $- 3 x^{3} + 2 x^{2} + 1$ $\neq$ $- 3 x^{3} - 2 x^{2} - 1$ , the function is not odd.

The function is not odd

Step 5

The function is neither odd nor even

Step 6