Question

factor out the greatest common factor in the given polynomial.
3(8y - 9)² + 7(8y - 9)³

3(8y - 9)² + 7(8y - 9)³ = \square
(factor completely.)

Explanation:

Step1: Identify the GCF

The terms are \(3(8y - 9)^2\) and \(7(8y - 9)^3\). The greatest common factor (GCF) of the two terms is \((8y - 9)^2\) (since it's the lowest power of the common binomial factor).

Step2: Factor out the GCF

Factor out \((8y - 9)^2\) from each term:
\[

$$\begin{align*} &3(8y - 9)^2 + 7(8y - 9)^3\\ =&(8y - 9)^2[3 + 7(8y - 9)] \end{align*}$$

\]

Step3: Simplify the bracket

Simplify \(3 + 7(8y - 9)\):
\[

$$\begin{align*} 3 + 7(8y - 9)&=3 + 56y - 63\\ &=56y - 60 \end{align*}$$

\]

Step4: Factor out common factor from bracket

Factor out 4 from \(56y - 60\): \(56y - 60 = 4(14y - 15)\)
So putting it all together: \((8y - 9)^2\times4(14y - 15)=4(8y - 9)^2(14y - 15)\)

Answer:

\(4(8y - 9)^2(14y - 15)\)