Question

In the following exercises, factor using substitution${\left(3y-2\right)}^2-\left(3y-2\right)-2$

Short answer

The factors are$\left(3y-4\right)\left(3y-1\right)$

Step-by-step solution

Step 1. Given Information

The trinomial is${\left(3y-2\right)}^2-\left(3y-2\right)-2$

Find factors using substitution method.

Step 2. Find the term for substitution

Rewrite the trinomial to prepare for the substitution

${\left(3y-2\right)}^2-\left(3y-2\right)-2$

Substitute$3y-2=u$

So, trinomial becomes

$u^2-u-2$

Step 3. Factor the trinomial

Split the middle term of trinomial to find factors

$$\begin{gathered} u^2-u-2 \\ {u}^2-2u+u-2 \end{gathered}$$

Factor it

$$\begin{gathered} u\left(u-2\right)+1\left(u-2\right) \\ \left(u-2\right)\left(u+1\right) \end{gathered}$$

Step 4. Substitute back u

Substitute back $u=3y-2$

$$\begin{gathered} \left(3y-2-2\right)\left(3y-2+1\right) \\ \left(3y-4\right)\left(3y-1\right) \end{gathered}$$