Q. 128E - Intermediate Algebra

Question

In the following exercises, factor using substitution. 

$x^4+2x^2-8$

Step-by-Step Solution

Answer

The factors are $\left(x^2+4\right)\left(x^2-2\right)$

Step 1. Given Information

The trinomial is $x^4+2x^2-8$.

Find factors using substitution method. 

Step 2. Find the term for substitution

Rewrite the trinomial to prepare for the substitution  ${\left(x^2\right)}^2+2x^2-8$

Substitute $x^2=u$

So, trinomial becomes $u^2+2u-8$

Step 3. Factor the trinomial

Split the middle term of trinomial to find factors

 $u^2-2u+4u-8$

Factor it

$u\left(u-2\right)+4\left(u-2\right)$

$\left(u-2\right)\left(u+4\right)$

Step 4. Substitute back u

Substitute back $u=x^2$

$\left(x^2+4\right)\left(x^2-2\right)$

Step 5. Check

Multiply all factors 

$\left(x^2+4\right)\left(x^2-2\right)$

$$\begin{gathered} x^4+4x^2-2x^2-8 \\ {x}^4+2x^2-8 \end{gathered}$$