Question

You were asked to factor the following expression by grouping: 30uv+30u−36u ² −25v.

A) 6u(5v+5−6u−5)
B) 5(6u−5)(v+6u+5)
C) 5u(6v+6−5u−5)
D) 5(6u−5)(6u+5−v)

Answer 1

The expression 30uv+30u−36u²−25v is correctly factored by grouping as 5(6u−5)(v−6u), corresponding to option D.

Step-by-step explanation:

The student was asked to factor the following expression by grouping: 30uv+30u−36u²−25v. The correct approach to factor by grouping is to first group the terms in pairs and then to find the common factors within each group. In this case, we will group the terms as (30uv+30u) and (−36u²−25v).

First Group: 30uv+30u = 30u(v+1)

Second Group: −36u²−25v = −6u(6u+5v)

Now, we notice that we have made a mistake since the second group does not reveal a common factor with the first group. Let's regroup and factor out a negative sign in the second group to see if that helps.

Regrouping: 30uv+30u+(-36u²)−25v = 30u(v+1)−6u(6u+5) because factoring − from −36u² and −25v gives us −6u(6u+5).

Now, we have a common factor of (6u+5) in both groups. We can factor this out:

30u(v+1)−6u(6u+5) = 5(6u+5)(v+1)−(6u+5) = 5(6u+5)(v+1−1) = 5(6u−5)(v−6u)

So, the factored form of the expression by grouping is 5(6u−5)(v−6u), which corresponds to option D.