Question

In the following exercises, factor using substitution$x^4-3x^2-2$

Short answer

The factors are$\left(x^2-7\right)\left(x^2+4\right)$

Step-by-step solution

Step 1. Given Information

The trinomial is$x^4-3x^2-2$

Find factors using substitution method.

Step 2. Find the term for substitution

Rewrite the trinomial to prepare for the substitution

${\left(x^2\right)}^2-3x^2-28$

Substitute $x^2=u$

So, trinomial becomes$u^2-3u-28$

Step 3. Factor the trinomial

Split the middle term of trinomial to find factors

$u^2-7u+4u-28$

Factor it

$$\begin{gathered} u\left(u-7\right)+4\left(u-7\right) \\ \left(u-7\right)\left(u+4\right) \end{gathered}$$

Step 4. Substitute back u

Substitute back $u=x^2$

$\left(x^2-7\right)\left(x^2+4\right)$

Step 5. Check

Multiply all factors

$$\begin{gathered} \left(x^2-7\right)\left(x^2+4\right) \\ {x}^4-7x^2+4x^2-28 \\ {x}^4-3x^2-28 \end{gathered}$$