To factor the expression [tex]\(6v^2 + 2y - vy - 12v\)[/tex] by grouping, follow these detailed steps:
1. Group the terms in pairs:
[tex]\[
(6v^2 - 12v) + (2y - vy)
\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- From the first group [tex]\(6v^2 - 12v\)[/tex], factor out [tex]\(6v\)[/tex]:
[tex]\[
6v(v - 2)
\][/tex]
- From the second group [tex]\(2y - vy\)[/tex], factor out [tex]\(y\)[/tex]:
[tex]\[
2(y - \frac{vy}{2}) \quad \text{or factor out} \quad -(\text{GCF}) \rightarrow y(-v + 2)
\][/tex]
3. Rewrite the expression with these factored groups:
[tex]\[
6v(v - 2) - y(v - 2)
\][/tex]
4. Now, factor out the common binomial factor [tex]\((v - 2)\)[/tex]:
[tex]\[
(v - 2)(6v - y)
\][/tex]
5. Consolidate the factors:
- Here we notice that we can rewrite [tex]\(6v - y\)[/tex] with the common factor pulled out appropriately, which gives our final factorized form.
So, the fully factored form of the expression [tex]\(6v^2 + 2y - vy - 12v\)[/tex] using grouping is:
[tex]\[
-(v - 2)(-6v + y)
\][/tex]
Thus, the factorized form of the expression is:
[tex]\[
-(-6v + y)(v - 2)
\][/tex]
