# Factoring Trinomials with a Negative Constant

## Factoring Trinomials with a Negative Constant

Factor x² plus 6x, minus 27. We’re trying to factor this trinomial into two binomials. First of all, these first terms in each binomial have a product of x², so you have to think, “What would you multiply together to get a product of x²?” That’s x times x, x times x is x².

Now to find these constants in your binomial you’re going to have to find the factors of your last term, which in this case is a negative 27, that have a sum of your middle term which is a positive 6. Here’s where it gets a little bit tricky, the factors of a negative number are a positive number and a negative number, so we know then that we’re going to multiply a positive number times a negative number to get this negative product.

Which means you’ve, like, doubled the number of factors for negative 27 because you have positive 1 times negative 27 or negative 1 times positive 27, and so on and so forth. To pare that down, look at the sum that you’re trying to get; you want a positive sum.

In order to add a positive number and a negative number together and get a positive sum, you’re positive number needs to be the larger number, so I like to circle this so that I know my bigger number needs to be my positive number, so instead of negative 1 and positive 27 and then positive 1 and negative 27, I now know that the 27 has to be the positive factor because that way when I add these together I get a positive sum.

Of course, it’s not 6 it’s 26, so we have to keep going. The next factor for negative 27 would be negative 3 and positive 9. Negative 3 times 9 is negative 27, negative 3 plus 9 is positive 6, so those are our factors, a negative 3 and a positive 9, and there we’ve factored it.

Let’s look at another example. Again, we’re factoring this into two binomials. X times x is x², and now we need to find the factors of our last term, our constant, negative 14 that have a sum of our middle coefficient, negative 5.

Again, to get a negative product like negative 14, we have to multiply a positive number times a negative number. This time, we want that positive and negative number to have a sum that’s negative, which means my negative number is going to be my larger number this time. 1 and negative 14, that has a product of negative 14 and a sum that’s negative, it’s just not negative 5, it’s negative 13.

Positive 2 and negative 7, 2 times negative 7 is negative 14, and 2 plus negative 7 is negative 5, so those are our factors, plus 2 and minus 7. Now, keep in mind that every time you factor you can check your answer by multiplying those binomials back together.

If your product is equal to your original polynomial, then you factored correctly. Let’s practice that on just the second example. I’m going to FOIL these, which means “First, Outer, Inner, Last;” it’s just a way to organize your multiplication.

First, x times x is x². Outer, x times negative 7, negative 7x. Inner, 2 times x. Last, 2 times negative 7. Now combine these like terms, and you get x² minus 5x, minus 14. Since that matches what we were factoring, we know we factored correctly.