Sure, I'll guide you through the steps to factor the given expression by grouping.
We start with the expression:
[tex]\[ 8v + vu - 4v^2 - 2u. \][/tex]
To factor this by grouping, follow these steps:
1. Group the terms in pairs:
[tex]\[ 8v + vu - 4v^2 - 2u. \][/tex]
We can group them as follows:
[tex]\[ (8v + vu) - (4v^2 + 2u). \][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- From the first group, [tex]\(8v + vu\)[/tex], the GCF is [tex]\(v\)[/tex]:
[tex]\[ 8v + vu = v(8 + u). \][/tex]
- From the second group, [tex]\(4v^2 + 2u\)[/tex], the GCF is [tex]\(2\)[/tex]:
[tex]\[ 4v^2 + 2u = 2(2v^2 + u). \][/tex]
Notice that we actually need to maintain the sign consistent while factoring, so we factor out [tex]\(-2\)[/tex] (because both terms are opposite in sign in the original expression):
[tex]\[ - (4v^2 + 2u) = -2(2v^2 + u). \][/tex]
Now we have:
[tex]\[ v(8 + u) - 2(2v^2 + u). \][/tex]
3. Look for a common binomial factor:
It seems there's an error in our grouping approach as it should give us a simpler factorable form. Let’s try grouping differently or another way:
Split and regroup terms to align for easier common factorization:
[tex]\[ 8v + vu - 4v^2 - 2u = 8v - 4v^2 + vu - 2u. \][/tex]
Group them as:
[tex]\[ (8v - 4v^2) + (vu - 2u). \][/tex]
4. Factor out the GCF from each group again:
- From [tex]\(8v - 4v^2\)[/tex], the GCF is [tex]\(4v\)[/tex]:
[tex]\[ 8v - 4v^2 = 4v(2 - v). \][/tex]
- From [tex]\(vu - 2u\)[/tex], the GCF is [tex]\(u\)[/tex]:
[tex]\[ vu - 2u = u(v - 2). \][/tex]
Now we have:
[tex]\[ 4v(2 - v) + u(v - 2). \][/tex]
5. Recognize the common factors (factoring involves noting a rearrangement and sign consistency):
Rewrite both groups:
[tex]\[ 4v(2 - v) + u(v - 2) \][/tex]
Align terms carefully noticing:
[tex]\[ 4v(2 - v) - u(2 - v) \text{, which simplifies factoring similar binomial.}\][/tex]
6. Simplify the expression:
Note the equivalent factor through commutation:
[tex]\[ (4v - u)(2 - v). \][/tex]
So the final factored form of the expression:
[tex]\[ 8v + vu - 4v^2 - 2u \][/tex]
is:
[tex]\[ (u - 4v)(v - 2). \][/tex]
