Question

Factor 3x^2y^2 − 2xy^2 − 8y^2. Show your work.

Answer 1

1) Look for common factors. You see that y^2 is a factor of every term so you can remove it to get

... = (y^2)(3x^2 -2x -8)

The quadratic in x can be factored by your favorite method. There is one called by various names that has you look for factors of (3)(-8) that add to (-2). When the quadratic is written as ax^2+bx+c, you're looking for factors of the product "ac" that add to "b". Of course, you know that

... -24 = -24*1 = -12*2 = -8*3 = -6*4

the last factor pair shown here has a sum of -2, so our factorization is

... = (y^2)(3x -6)(3x +4)/3 . . . . . the "a" coefficient is repeated in each factor (at first), then divided out

... = (y^2) (x -2) (3x +4)

2) You recognize this expression to be of the form

... (x +a)^2 = x^2 +2ax + a^2

where a=5. As a result, you know the factorization is

... = (x +5)^2

3) You recognize this expression to be the difference of squares, so you know the factorization is

... a^2 - b^2 = (a -b)(a +b)

where a=x and b=6. As a result, you know the factorizatin is

... = (x -6) (x +6)