How do I get the answer to Question: Factor 3x^2+5x-2

Question

I know the answer: (x+2)(3x-1)

But I want to know HOW to get this answer

Answer 1

Hi Vince,

 

I'll be happy to show you how to factor a trinomial when the leading coefficient is greater than 1.

 

Let's begin by writing the standard form of a quadratic equation and use it for reference in our explanation:

    ax² + bx + c = 0

 

The procedure will be as follows:

    1. Expand the trinomial into 4 terms by rewriting

        coefficient (b) as the sum of two factors.  These

        two factors will come from the product of

        coefficients (a) and (c).

    2.  Distribute the two factors over x, which will

        expand the middle term of the trinomial into two

        terms, which in turn will expand the trinomial

        into 4 terms.

    3.  Now we will group these 4 terms into two 

         factorable binomial pairs.

    4.  Finally, we'll factor each binomial pair and pull

         out the common factor from each factored

         binomial pair and we're done !!

 

Let's apply the above procedure to our problem:

    3x² + 5x - 2

    a = 3; b = 5; c = -2

    product of coefficients (a) and (c) = (3)(-2) = -6

    factors of -6 which add up to b:  6 - 1 = 5

    rewrite (b) as sum of (6 and -1):

        3x² + (6 - 1)x - 2

    expand middle term by distributing x:

        3x² + 6x -x - 2

    group terms for factoring:

        (3x² + 6x) and (-x - 2)

    factor each binomial pair:

        3x(x + 2) and -1(x + 2)

    pull out common factor from each factored pair:

        (x + 2)(3x - 1) --->  voila !!

 

This method of factoring is nicknamed the "bomb" method, since the middle term of the trinomial is "blown up" into two terms !!

 

Well, there we have it - no more guess work needed - just a nice orderly procedure to get the job done.

 

Enjoy.

 

God bless, Jordan.