SOLUTION: Hi, can you please help me? I need to factor the following completely: 21x^3-18x^2y+24xy^2 Can you please show me how? thank you!

Question 403073: Hi, can you please help me?
I need to factor the following completely:
21x^3-18x^2y+24xy^2
Can you please show me how?
thank you!
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

When factoring, always start with the Greatest Common Factor (GCF). The GCF here is 3x:

Always keep factoring until no more factoring can be done. The second factor is a trinomial we should try to factor. It does not fit any of the patterns for factoring. But we can try trinomial factoring. This factoring is "un-FOIL-ing" the expression. We are looking to see if some pair of binomials will result in when multiplied. We are looking for

where and (the "First" terms) are factors of and and (the "Last" terms) are factors of . That is the easy part the hard part is that the "Outside" terms, and , and "Inside" terms, and must produce the -6xy if the expression is to factor at all.

For and there is really only one choice: 7x and x. For and there are several choices. Since the middle term has a minus in front of it we need them both to be negative. But they could be -8y and -y or -2y and -4y. And which one is and which one is is also not known yet.

So the possibilities are:
(7x-8y)(x-y)
(7x-y)(x-8y)
(7x-2y)(x-4y)
or
(7x-4y)(x-2y)
All of these will produce the and the . But only 1 (or none) of these will produce the -6xy. And that will come from the Outside and Inside terms. Let's try each one, looking at just the Outside and Inside multiplications of FOIL:
(7x-8y)(x-y)
Outside: 7x*(-y) = -7xy
Inside: x*(-8y) = -8xy
Together they make -15xy

(7x-y)(x-8y)
Outside: 7x*(-8y) = -56xy
Inside: x*(-y) = -xy
Together they make -57xy

(7x-2y)(x-4y)
Outside: 7x*(-4y) = -28xy
Inside: x*(-2y) = -2xy
Together they make -30xy

(7x-4y)(x-2y)
Outside: 7x*(-2y) = -14xy
Inside: x*(-4y) = -4xy
Together they make -18xy

None of these produced the -6xy we were looking for. So will not factor any further. So your fully factored expression is:

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