Question

Which of the following shows the factors of 9x^2+3x-2?
(9x-2)(x+1) is this correct?

Answer 1

No, (9x-2)(x+1) factors out to 9x^2+7x-2

This one doesn't look like it would factor

Answer 2

You can test your factoring by multiplying it out:
\[(9x-2)(x+1) = 9x*x + 9x*1 -2*x -2*1 = 9x^2+7x-2\]That is not what you factored, so it is not correct.

It is possible to factor this expression, however:
\[9x^2+3x-2\]There aren't any common factors that we can factor out. However, we can factor the \(9x^2\) term in a few ways:

\(3x*3x\) is an obvious choice, let's see if it can be made to work:

\[(3x + a)(3x + b) = 9x^2 +3x- 2\]If we expand our left hand side:
\[(3x+a)(3x+b) = 3x*3x + 3x*b + a*3x + a*b\]\[9x^2 + (3a+3b)x + ab\]

Compare that with our original:
\[9x^2 + (3a+3b)x + ab = 9x^2 + 3x - 2\]If we can pick values of \(a,b\) that make \((3a+3b) = 3\) and \((ab) = -2\) then we are done. Easier to crack the \(ab = -2\), I think. Our choices are -1*2 and -2*1. Don't know which yet, but let's try those two in the second one:\[(3a+3b) =3\]
\[3(-2)+3(1) = -6+3 = -3\]okay, \(a=-2,b=1\) is not it

\[3(-1) + 3(2) = 3\]\[-3+6 = 3\]So that works. Plug \(a=-1,\ b=2\) back into our prototype factored equation:
\[(3x+a)(3x+b) =(3x-1)(3x+2)\]

Let's multiply to make sure it is correct:
\[(3x-1)(3x+2 = 3x*3x + 3x*2 -1*3x -1*2 = 9x^2 +6x-3x-2\]\[ = 9x^2+3x-2\checkmark\]

Answer 3

Thank you!