Math CentralQuandaries & Queries


Question from Ema, a student:

How do you factor this completely. x^5y^3-3x^4y^2-28x^3y

Hi Ema.

The technical name for that kind of algebraic expression is a bivariate polynomial.

I'm going to do a similar problem to yours to show you the technique.

Factor completely:
4a7b3 - 10a6b2 - 24a5b.

All the terms are even, so I can factor out a 2.
All the terms have at least a5 in them.
All the terms have at least b in them.
So factor out a 2a5b first.

2a5b (2a2b2 - 5ab - 12)

Now I can see that what is in the parentheses is in the form of Ax2 + Bx + C, where in this case x = ab and A, B, and C are all constants. So this is actually a normal trinomial like you've solved before. In this case, you have 2x2 - 5x - 12. Use normal decomposition factoring to break this down. If you aren't sure about that, Penny wrote a good description here. You get (2x + 3)(x - 4). Then we can replace x with ab.

Since 2a2b2 - 5ab - 12 = (2ab + 3)(ab - 4), my final answer is 2a5b(2ab + 3)(ab - 4).

Hope this helps,
Stephen La Rocque.

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