Q. 131. - Intermediate Algebra

Question

In the following exercises, factor using substitution ${(x-3)}^2-5(x-3)-36$

Step-by-Step Solution

Answer

The factors are $\left(x-12\right)\left(x+1\right)$

Step 1. Given information

The trinomial is ${(x-3)}^2-5(x-3)-36$.

Find factors using substitution method. 

Step 2. Find the term for substitution

Rewrite the trinomial to prepare for the substitution  

${(x-3)}^2-5(x - 3)-36$

Substitute binomial $\left(x-3\right)=u$

So, trinomial becomes 

$u^2-5u-36$

Step 3. Factor the trinomial

Split the middle term of trinomial to find factors

$$\begin{gathered} u^2-5u-36 \\ {u}^2-9u+4u-36 \end{gathered}$$

Factor it

$$\begin{gathered} u\left(u-9\right)+4\left(u-9\right) \\ \left(u-9\right)\left(u+4\right) \end{gathered}$$

Step 4. Substitute back u

Substitute back $u=x-3$

$$\begin{gathered} \left(x-3-9\right)\left(x-3+4\right) \\ \left(x-12\right)\left(x+1\right) \end{gathered}$$