Factor completely

Question

Factor completely 4x^2 + 8x - 96

Answer 1

Factor completely 4x^2 + 8x - 96

The first thing to notice is that 4 is a factor:

4x² + 8x - 96 = 4(x² + 2x - 24).

To factor the quadratic polynomial in the parentheses (notice the coefficients a=+1, b=+2 and c=-24), you need to find two numbers, x₁ and x₂, whose product = c = -24 where the negative of their sum = b = +2. Those two numbers, which you can guess and check, are 4 and -6.

The binomial factors of the polynomial in the parentheses will be

x² + 2x - 24 = (x-x₁)(x-x₂) = (x-4)(x+6).

Alternatively, if you don't like to guess and check, you can use the quadratic formula to calculate

x₁ = (-2 + (2² - 4(-24) )^1/2)/2 = (-2 + (4 + 96)^1/2)/2 = (-2 + 10)/2 = 4 and

x₂ = (-2 - (2² - 4(-24) )^1/2)/2 = (-2 - (4 + 96)^1/2)/2 = (-2 - 10)/2 = -6

but I consider that overkill!

So the answer is:

4x² + 8x - 96 = 4(x-4)(x+6).

Answer 2

Step 1) Look for common factors (something that is in all terms)

In this case each term is divisible by 4 so we can factor out a 4:

4(x² + 2x - 24)

Step 2) Factor into binomials if possible.

There are lots of ways to do this. There are tools like the diamond and box method you can use (you can find youtube videos on how to do these). I like to do this:

Write down the parenthesis to give you a place to start:

4( )( ) now fill in as much as you can. You know the first terms in both binomials must be "x" because "x" times "x" is "x²" so fill in those first terms:

4(x )(x ). Now, thing of all the factors of the last term (24). They are:

1, 24

2, 12

3, 8

4, 6

and look through that list until you find two numbers that are 2 apart (2 because the middle term is 2). I see 4 and 6 are 2 apart. In order for them to multiply to get negative 24, one of those numbers must be a positive and one must be a negative. Since the middle term is positive, we want to assign the positive to the larger number, 6. So:

4(x + 6)(x - 4).

Step 3) Check your answer

Use the FOIL method to multiply out the binomials to make sure they give you the original problem. FOIL stands for First, Outside, Inside, Last where you multiply the two First terms together, then multiply the two Outside terms and add to the previous, then multiply the two Inside terms and add to the previous, then multiply the two Last terms and add them to the previous. Doing so give us the original answer so we are good.

Answer 3

4x² + 8x - 96 = 4(x² + 2x - 24) = 4(x - 4)(x + 6)

Test the answer.