Q. 381 - Intermediate Algebra

Question

In the following exercise, factor using substitution.

$x^4-13x^2-30$

Step-by-Step Solution

Answer

Thus, the factorisation of the given polynomial is :

$x^4-13x^2-30=(x^2-15)(x^2+2)$

Step 1. Given information

The given polynomial is:

$x^4-13x^2-30$

Step 2. substitute x 2 as u

Substituting $x^2=u$ in the given polynomial, we get:

$u^2-13u-30$

Step 3. Perform factorisation

Factorising the polynomial $u^2-13u-30$, we get:

$$\begin{gathered} u^2-13u-30 \\ {u}^2-15u+2u-30 \\ u(u-15)+2(u-15) [taking out u and 2 as common] \\ (u+2)(u-15) [taking (u-15) as a common] \end{gathered}$$

Now, again substituting $u=x^2$, we get:

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

Thus, the factorisation of the given polynomial is:

$x^4-13x^2-30=(x^2-15)(x^2+2)$

Step 4. Check the answer

Multiplying the obtained factors we get:

$$\begin{gathered} x^4-13x^2-30=\left(x^2+2\right)(x^2-15) \\ {x}^4-13x^2-30=x^4-15x^2+2x^2-30 \\ {x}^4-13x^2-30=x^4-13x^2-30 \end{gathered}$$

This is true.

So the factors are correct.