Question

factor the polynomial expression ( x^4 + x^2 - 20 ). drag each expression to the correct location on the factored solution. each expression can be used more than o all expressions will be used. ( x^4 + x^2 - 20 = (x^2 + square) (square + square) (square - square) ) expressions: ( x ), ( x^2 ), ( 4 ), ( 2 ), ( 5 ), ( 10 ), ( 1 )

Explanation:

Step1: Substitute $u=x^2$

Let $u = x^2$, so the polynomial becomes $u^2 + u - 20$.

Step2: Factor quadratic in $u$

Find two numbers that multiply to $-20$ and add to $1$: $5$ and $-4$.

$$\begin{align*} u^2 + u - 20 &= (u + 5)(u - 4) \end{align*}$$

Step3: Substitute back $u=x^2$

$$\begin{align*} (u + 5)(u - 4) &= (x^2 + 5)(x^2 - 4) \end{align*}$$

Step4: Factor difference of squares

$x^2 - 4$ is a difference of squares: $x^2 - 2^2 = (x + 2)(x - 2)$.

$$\begin{align*} (x^2 + 5)(x^2 - 4) &= (x^2 + 5)(x + 2)(x - 2) \end{align*}$$

Answer:

$x^4 + x^2 - 20 = (x^2 + 5)(x + 2)(x - 2)$
(Blanks filled as: 5, $x$, 2, $x$, 2)