Question

How do you complete the factorization of `3x ^ { 2} - 10x y - 48y ^ { 2}`?

Answer 1

Please observe that `3x^2` can only factor into `(3x)(x)`.

If the polynomial factors evenly, then a good guess for the general form is:

`(3x+ay)(x-by)`

NOTE: If the guess is incorrect and the general form is `(3x-ay)(x+by)`, then we will find that a and b are negative and we will change the guess.

Matching the middle terms:

`axy - 3bxy = -10xy`

Divide both sides by xy and mark as equation [1]:

`a - 3b = -10" [1]"`

Matching the end terms:

`-aby^2 = -48y^2`

Divide both sides by `-y^2` and mark as equation [2]:

`ab = 48" [2]"`

We can use equations [1] and [2] to solve for a and b.

Write equation [1] as:

`a = 3b-10" [1.1]"`

Substitute into equation [2]:

`(3b-10)b = 48`

Multiply:

`-3b^2-10b = 48`

Write in standard form:

`3b^2-10b - 48 = 0`

factor

`(3b+8)(b-6) = 0`

`b = -8/3` and `b = 6`

One of these is extraneous. Use equation [1.1] to find the corresponding values of a:

`a = 3(-8/3)-10` and `a = 3(6)-10`

`a = -18` and `a = 8`

Check the integer solution `a= 8 and b = 6`:

`(3x+8y)(x-6y) = 3x^2-18xy + 8xy -48y^2`

`(3x+8y)(x-6y) = 3x^2-10xy -48y^2`

This checks. The other solution is extraneous.

The factors are:

`3x^2-10xy -48y^2 = (3x+8y)(x-6y)`