How do you factor a trinomial when a =/= 1?

Question

I didn't get this in class. Here's a problem. It's a practice one in the book and I don't know how they got the answer. Could you explain it to me?

4x^2 - 4x - 15

Answer 1

I can try, sure.

What you're going to do is trial and error; you're going to split 4x^2 into two (either 2x and 2x, or 4x and x) and then test the outer and inner to see if they add up to the coefficient of the middle term.

(4x + ?) (x + ?)

For the question marks, what we want to test are factors of -15. Let's try -3 and 5.

(4x - 3) (x + 5). The outer terms multiplied is 20x, the inner terms multiplied is -3x. 20x + (-3x) = 17x, which is NOT our middle term.

Try -15 and 1.

(4x - 15) (x + 1). Outer = 4, inner is -15, outer + inner = -11. This isn't the middle coefficient. Scrap the idea of choosing 4x and x altogether, and try (2x + ?) (2x + ?)

Try -3 and 5

(2x - 3) (2x + 5)
Outer = 10x, inner = -6x. Outer + inner = 4x. Is this our middle term? CLOSE; we want -4x, NOT +4x. Whenever we get a close answer such as this, all we have to do is flip the signs; that is, we try 3 and -5, and it should automatically work. Let's see for ourselves.

(2x + 3) (2x - 5).
Outer + inner = -10x + 6x = -4x. SCORE!!!

Therefore, our factorization is
(2x + 3) (2x - 5)

Answer 2

These problems are solved by trial and error. A few hints..


Beause there are two minus signs in this trinomial, you know that the form of the equation has to be

(ax-b) * (cx+d) (1)

That is, if both terms had + signs, then the trinomial would all have plus signs (since +b * +d = +bd), and if both terms had minus signs, then the last term of the trinomial would be positive (since -b * -d = +bd). The only way to get a trinomial where the last two terms are negative is for the one factored term to be positive and one to be negative.

Step 2:


Now for the trial and error part.

The form in equation 1 shows that the first terms in each factor, when multiplied together has to be the following:
ax * cx = 4x^2

There are several possibilities for a and c:
a=4, c=1
a=1, c=4
a=2, c=2

a=-4, c=-1
a=-1, c=-4,
a=-2, c=-2


Also, you know that -b * +d = -15.

There are only two possibilities here
b=-3, d=5
b=3, d=-5

So, try the various combinations of a, c, b, d until you get the result to work out. Factoring comes with practice; initially it takes a lot of trial and error to get it to work out, but eventually you will be able to solve these almost by inspection.

Keep practicing!

-Guru

Answer 3

First, figure the factors of 4x^2= 4x * x or 2x * 2x
Now, the factors of 15 = 1 x 15, 3, x 5

Now we have to guess which combination create the -4x in the middle position.

(4x - 5)(x+3) = 4x^2+12x-5x-15 = 4x^2+7x-15, so not this one
(4x-3)(x+5) = 4x^2+20x-3x-15 = 4x^+17x-15, so not this one
(2x-5)(2x+3) = 4x^2 +6x -10x-15 = 4x^2-4x-15

So your answer is (2x-5)(2x+3)

Answer 4

It would be (2x+3)(2x-5) because it works the same way as a trinomial with a leading coefficient of 1. You just try out the different factors of the leading coefficient.

Answer 5

These sites might help you.

Factoring A Trinomial
http://www.algebrahelp.com/lessons/factoring/trinomial/

GCF and Factoring Trinomials
http://www.ltcconline.net/greenl/courses/152B/FactoringRatExpr/factor.htm

Hints for solving the problems
Recognizing patterns
http://fp.academic.venturacollege.edu/rbrunner/hints_7.3.htm

http://www.learner.org/channel/workshops/algebra/workshop5/lessonplan1b.html