factor by grouping polynomials

[bobisaka]

p² + 5p + pq + 5q

(p +5)(p+q)

p²+pq+5p+5q

I am using the FOIL method for the factored polynomial, did i answer it correct?
Should i redistribute the monomials first, so the answer is the same?

 

[Dr.Peterson]

Looks correct to me, if the first line is the given polynomial, the second line is your factored form, and the third line is you check (by multiplying). Since the last line equals the first, it is correct. (Order doesn't matter - remember the commutative property.)

Why are some terms underlined? What do you mean by "redistribute"?

 

[Otis]

… Should i redistribute the monomials … so [my result] is the same?

Hello bobisaka. You're asking about changing pq+5p to 5p+pq , yes? We are free to do that. Dr. Peterson told you why that order doesn't matter: The Commutative Property of Addition.

pq + 5p = 5p + pq

Therefore: p^2 + pq + 5p + 5q = p^2 + 5p + pq + 5q

?

 

[bobisaka]

Looks correct to me, if the first line is the given polynomial, the second line is your factored form, and the third line is you check (by multiplying). Since the last line equals the first, it is correct. (Order doesn't matter - remember the commutative property.)

Why are some terms underlined? What do you mean by "redistribute"?

I meant order, not redistribute, just didn't have the correct term to use.
I underlined it because i thought both products had to be in the same order for the FOIL method to work on the factored form. But as you said, they don't because of commutative property.

There are so many terms and concepts to remember, which i will need to drill in my memory.

 

[Dr.Peterson]

Here's something that may help a little:

When I factor by grouping, I keep factors in their original order, like this:

p² + 5p + pq + 5q​

p(p + 5) + q(p + 5)​

(p + q)(p + 5)​

What was on the left (the p and q) stays on the left.

This doesn't matter at all, until I do the check. When I do FOIL (that is, multiply each term in the first binomial times each term in the second, in the natural order), the terms come out in their original order, which makes it just a tiny bit easier to check:

(p + q)(p + 5)​

p^2 + 5p + pq + 5q​

I don't have to search for matching terms! This way I can do it in my head more easily, if I want to.

You might also observe that if you swap the two middle terms and factor by grouping, you just end up with the same two factors in reverse order. After doing this a few times, you can get used to the fact that order doesn't matter in either direction.