Question

In the following exercises, factor using substitution.$x^2-13x^2-30$

Short answer

The factors are$\left(x^2+2\right)\left(x^2-15\right)$

Step-by-step solution

Step 1. Given Information

The trinomial is$x^4-13x^2-30$

Find factors using substitution method.

Step 2. Find the term for substitution

Rewrite the trinomial to prepare for the substitution

${\left(x^2\right)}^2-13x^2-30$

Substitute

$x^2=u$

So, trinomial becomes,

$u^2-13u-30$

Step 3. Factor the trinomial.

Split the middle term of trinomial to find factors

$u^2+2u-15u-30$

Factor it

$$\begin{gathered} u\left(u+2\right)-15\left(u+2\right) \\ \left(u+2\right)\left(u-15\right) \end{gathered}$$

Step 4. Substitute back u

Substitute back $u=x^2$

$\left(x^2+2\right)\left(x^2-15\right)$

Step 5. Check

Multiply all factors

$$\begin{gathered} \left(x^2+2\right)\left(x^2-15\right) \\ {x}^4+2x^2-15x^2-15 \\ {x}^4-13x^2-15 \end{gathered}$$