Determine if Odd, Even, or Neither x^4-x^10+1

$x^{4} - x^{10} + 1$

Step 1

Write $x^{4} - x^{10} + 1$ as a function.

$f (x) = x^{4} - x^{10} + 1$

Step 2

Find $f (- x)$ .

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

$f (- x) = {(- x)}^{4} - {(- x)}^{10} + 1$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{4} x^{4} - {(- x)}^{10} + 1$

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

$f (- x) = 1 x^{4} - {(- x)}^{10} + 1$

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

$f (- x) = x^{4} - {(- x)}^{10} + 1$

Apply the product rule to $- x$ .

$f (- x) = x^{4} - ({(- 1)}^{10} x^{10}) + 1$

Multiply $- 1$ by ${(- 1)}^{10}$ by adding the exponents.

Move ${(- 1)}^{10}$ .

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

Multiply ${(- 1)}^{10}$ by $- 1$ .

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

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

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

$f (- x) = x^{4} + {(- 1)}^{10 + 1} x^{10} + 1$

Add $10$ and $1$ .

$f (- x) = x^{4} + {(- 1)}^{11} x^{10} + 1$

Raise $- 1$ to the power of $11$ .

$f (- x) = x^{4} - x^{10} + 1$

Step 3

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

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

Since $x^{4} - x^{10} + 1 = x^{4} - x^{10} + 1$ , the function is even.

The function is even

Step 4