Q. 152E - Intermediate Algebra

Question

In the following exercises, factor each expression using any method $x^4-7x^2-8$

Step-by-Step Solution

Answer

The factors are $\left(x^2+1\right)\left(x^2-8\right)$

Step 1. Given Information

The trinomial is $x^4-7x^2-8$

Find factors using substitution method. 

Step 2. Find the term for substitution

Rewrite the trinomial to prepare for the substitution

${\left(x^2\right)}^2-7x^2-8$

Substitute $x^2=u$

So, trinomial becomes $u^2-7u-8$

Step 3. Factor the trinomial

Split the middle term of trinomial to find factors

$u^2+u-8u-8$

Factor it

$$\begin{gathered} u\left(u+1\right)-8\left(u+1\right) \\ \left(u+1\right)\left(u-8\right) \end{gathered}$$

Step 4. Substitute back u

Substitute back $u=x^2$

$\left(x^2+1\right)\left(x^2-8\right)$

Step 5. Check

Multiply all factors $\left(x^2+1\right)\left(x^2-8\right)$

$$\begin{gathered} x^4+x^2-8x^2-8 \\ {x}^4-7x^2-8 \end{gathered}$$