Expert: Abe Mantell - 7/18/2008

Question
I'm not understanding factoring. My instructor wants us to factor these problems completely and I am absolutely lost. I feel as though she is speaking a foreign language. Can you help? If so, I thank you in advance.  x^10y^3 - 4x^9y^2 - 21x^8y   next problem    x^3 - 2x^2 + 7x - 14    next problem 7x^2 + 58x + 16    any help at all would be appreciated. They do not teach the idiot edition of Algebra and I am discouraged as I get A's in all other classes but Algebra seems beyond my comprehension.

Answer
Hello Angela,

The first thing to look for when factoring is common factors.
1.  x^10 y^3 - 4x^9 y^2 - 21x^8 y, notice each term has at least
-- an x^8 and y factor, so we factor those out, giving:
-- x^8 y (x^2 y^2 - 4xy - 21) = x^8 y [(xy)^2 -4 (xy) - 21]
-- Let W=xy, so [(xy)^2 -4 (xy) - 21] looks like W^2 - 4W - 21,
-- which factors (W-7)(W+3), now put back xy for W, so the final
-- factored form is: x^8 y (xy-7)(xy+3)...OK?

2. x^3 - 2x^2 + 7x - 14 has no common factor, but we can factor
-- "by grouping"...the firs two terms have an x^2 common factor,
-- so we can write x^3 - 2x^2 as x^2 (x-2), the next two terms
-- have a "7" in common, so we can write 7x - 14 as 7(x-2).
-- so, we have x^3 - 2x^2 + 7x - 14 = x^2 (x-2) + 7 (x-2), now
-- notice these two both have an (x-2) in common!  Factoring that out
-- gives: (x-2)(x^2 + 7)...OK?

3. 7x^2 + 58x + 16 = (7x + 2)(x + 8)

I hope this helps...

Abe