I need help with factoring polynomials using pi and sigma. I understand it, but what is the best way to find the sigma when the terms are, in my opinion, really large? It takes me forever to find it, guessing.

Question

Let's say the problem is this:
56x² - 53xy + 12y²
The pi = 672 = (-____)(-_____)
and the sigma = -53 = (-__) + (-___)
How would you solve the problem?

Answer 1

If the method is not convenient then you should not use.

The factorization is of the form

A (x + p1 y) (x + p2 y)

with A = 56.

If you equate the polynomial to zero and solve for x, the solutions are:

x = -p1 y and x = -p2 y

So, we substitute x = -p y in the polynomial and equate it to zero:

(56 p^2 + 53 p + 12) y^2 = 0

We know that if p = p1 or p2, this will be zero for nonzero y.

So, p1 and p2 are the solutions of the equation:

56 p^2 + 53 p + 12 = 0

p = (-53 +-11)/(2*56)