Q. 151E - Intermediate Algebra

Question

In the following exercises, factor each expression using any method $x^4-4x^2-12$

Step-by-Step Solution

Answer

The factors are $\left(x^2+2\right)\left(x^2-6\right)$

Step 1. Given Information

The trinomial is $x^4-4x^2-12$

Find factors using substitution method. 

Step 2. Find the term for substitution

Rewrite the trinomial to prepare for the substitution

${\left(x^2\right)}^2-4x^2-12$

Substitute $x^2=u$

So, trinomial becomes $u^2-4u-12$

Step 3. Factor the trinomial

Split the middle term of trinomial to find factors

$u^2+2u-6u-12$

Factor it

$$\begin{gathered} u\left(u+2\right)-6\left(u+2\right) \\ \left(u+2\right)\left(u-6\right) \end{gathered}$$

Step 4. Substitute back u

Substitute back $u=x^2$

$\left(x^2+2\right)\left(x^2-6\right)$

Step 5. Check

Multiply all factors $\left(x^2+2\right)\left(x^2-6\right)$

$$\begin{gathered} x^4+2x^2-6x^2-12 \\ {x}^4-4x^2-12 \end{gathered}$$