To factor the polynomial [tex]\( x^4 + x^2 - 20 \)[/tex], we begin by seeking expressions that multiply together to give us the original polynomial. The correct factorization is:
[tex]\[
x^4 + x^2 - 20 = (x - 2)(x + 2)(x^2 + 5)
\][/tex]
We can verify this by expanding the factors:
1. First, multiply [tex]\((x - 2)\)[/tex] and [tex]\((x + 2)\)[/tex]:
[tex]\[
(x - 2)(x + 2) = x^2 - 4
\][/tex]
2. Now multiply the result by [tex]\((x^2 + 5)\)[/tex]:
[tex]\[
(x^2 - 4)(x^2 + 5) = x^2 \cdot x^2 + x^2 \cdot 5 - 4 \cdot x^2 - 4 \cdot 5
\][/tex]
[tex]\[
= x^4 + 5x^2 - 4x^2 - 20
\][/tex]
[tex]\[
= x^4 + x^2 - 20
\][/tex]
So, the fully factored form of [tex]\( x^4 + x^2 - 20 \)[/tex] is indeed:
[tex]\[
(x - 2)(x + 2)(x^2 + 5)
\][/tex]
Therefore, the correct factors to place in the boxes are:
[tex]\[
x^4 + x^2 - 20 = (x - 2)(x + 2)(x^2 + 5)
\][/tex]
Let's place the correct expressions in the slots:
- (x - 2)
- (x + 2)
- (x^2 + 5)
The final factored form of the polynomial [tex]\( x^4 + x^2 - 20 \)[/tex]:
[tex]\[
x^4 + x^2 - 20 = (x - 2)(x + 2)(x^2 + 5)
\][/tex]
