Factorize 15r^2 - 31rt - 24t^2?

Question

Factorize 15r^2 - 31rt - 24t^2 and also explain.

By the way the book says the answere is (3r-8t)(5r+3t).

Thanks :]

Answer 1

First off, it's obviously going to be something along the lines of

(Ar + Bt)(Cr - Dt)

where A, B, C and D are constants.

Now:  Take the final coefficient and multiply it by the leading coefficient:

15(-24) = -360

We need factors of -360 that add up to -31.  Now we start looking:

-36 and 10:  -26, so we're close

-40 and 9:  -31, so that's it.

Now we write the trinomial as the sum of two binomials.  The order doesn't matter, but I'm going to arrange them to make our life a little easier:

15r^2 - 40rt + 9rt - 24t^2

Look at them in pairs and factor out common factors:

15r^2 - 40rt = 5r(3r - 8t)

9rt - 24t^2 = 3t(3r - 8t)

So, we have

15r^2 - 31rt - 24t^2 = 15r^2 - 40rt + 9rt - - 24t^2 =

5r(3r - 8t) + 3t(3r - 8t)

Note the common factor of (3r - 8t).  Factor it out and you're left with:

(3r - 8t)(5r + 3t)

and you're done.

This is a little difficult to explain in a medium like this, so I tried to put down every detail.  I hope it makes sense.