How would you answer this question? I am having trouble with it.

Question

Factoring Polynomials By Grouping

Can you group any two terms you want and still get the same answer? Show an example.

Answer 1

I would say yes and I can give 2 examples:

Example 1:

1. h^3 + h^2 - 6h - 6 can be grouped [ h^3 + h^2 ] - [ 6h + 6] then by factoring h^2 from first group and 6 from second group we get h^2(h+1) - 6(h+1) then the final factored form is (h^2 - 6) (h+1)
2. h^3 + h^2 - 6h - 6 can be grouped [ h^3 -6h] + [ h^2 - 6] then by factoring h from first group we get h(h^2 - 6) + (h^2 - 6) then the final factored form is (h + 1) (h^2 - 6)

Therefore, it didn't matter in which way we group the terms we got the same final factored form.

Example 2:

1. ab + 2a + 7b + 14 can be grouped as (ab + 2a) + (7b + 14) then by factoring out a from first group and 7 from second group we get a(b + 2) + 7(b + 2) then the final factored form is (a+7) (b + 2)
2. ab + 2a + 7b + 14 can be grouped as (ab + 7b) + (2a + 14) then by factoring out from the first group b and 2 from the second group we get b(a + 7) + 2(a+7) then the final form is (b + 2) (a+ 7)

Therefore, it didn't matter in which way we group the terms we got the same final factored form.

Answer 2

It depends on the number of variables in the problem. The key to factor by grouping is that you have to make sure that the combinations you arrange, or group together, can factor out. For example, say we have the polynomial 12a^3-9a^2+4a-3.

So here, we can group the 12a^3-9a^2 + 4a-3 and rewrite it as (12a^3-9a^2)+(4a-3). So we know that we just have to factor a 3 out of the first polynomial and we get 3a^2(4a-3)+1(4a-3). So when you factor it out, you will come out with (3a^2+1)*(4a-3). I hope this example makes sense to you. So in short the answer to your first question is no.