Factoring Trinomials with a Positive Constant
You have to ask yourself, “What would I multiply? What times what is x²?” Well that’s simply x times x, x times x is x². Now we have to figure out what the constants are in these binomials, and the constants are found by finding the factors of your last term, or of your constant, that have a sum of your middle coefficient.
The factors of 10 are 1, and 10, and 2, and 5, so out of these factors the factors that will also give you a sum of 7 are just the 2 and the 5, and they’re both positive because we have to get a product of 10 and a sum of 7, so a positive product means we have to multiply a positive number times a positive number or a negative number times a negative number, but since we need a positive sum then those numbers both have to be positive, so that’s plus 2 and plus 5.
That’s your answer, that is x² plus 7x, plus 10, factored, but you can always check it by FOILing your answer out. Let’s practice that. X times x, and again we’re FOILing, which stands for first times first, (so first term times first term) which is x².
Outer, these two terms on the outside, so x times 5, which is 5x. Inner is what the “I” stands for, so these are the two inside terms, 2 times x, which is 2x. Then, last, and these are the terms in the last position, 2 times 5 is 10.
Then we combine these like terms, which gives us x² plus 7x, plus 10, so you can see that we factored correctly because when we multiplied those binomials together our product was what we were factoring. Let’s look at another example. Factor x² minus 7x, plus 12.
Again, we know we’re looking for two binomials that we’re going to multiply together, it’s the two factors that will yield this product. That’s what we’re looking for, what factors will yield this product? First of all, our first term x² is found by multiplying x times x.
Now for our constants in our binomials, we need to find the factors of our constant the factors of our last term, 12, that have a sum of our middle coefficient, in this case it’s a negative 7, and that is important, we’re trying to find a sum that’s negative.
Well, the only way to get a negative sum is if at least one of your numbers is negative, but since we have a product that’s positive, or we’re trying to find the factors of a positive number, you can’t just have one negative factor to have a positive product.
That means that we’re going to be using only negative numbers, a negative times a negative is a positive, and then when you add a negative and a negative you get a sum that’s negative. The factors of 12. Negative 1 and negative 12, negative 1 times negative 12 is positive 12, and then if you add negative 1 and negative 12 you get a negative sum, it’s just not the right sum.
Let’s try negative 2 and negative 6, that has a sum of negative 8, so we’re still not there. Negative 3 and negative 4, negative 3 times negative 4 is positive 12, and negative 3 plus negative 4 is negative 7, so these are our two constants, this is x minus 3 and x minus 4.
Again, you can check this by FOILing it, or multiplying those two binomials together, to make sure that you get that product. First, x times x, x². Outer, x times negative 4, negative 4x. Inner, negative 3 times x, negative 3x. Last, negative 3 times negative 4, which is positive 12. Combine those like terms and you get x² minus 7x, plus 12, which is what we were trying to factor. There you have factoring trinomials with a positive constant.