Factoring Quadratics
The other really cool thing about quadratic polynomials is that we can factor them, usually pretty easily. When we say we're going to factor them, we're going to take the whole equation, f(x) = ax2 + bx + c, and we're going to find two things that multiply together to make that whole. So, for example, 2x2 + 7x + 3 can be written as (2x + 1)(x + 3). We have factored 2x2 + 7x + 3 into two things that can be multiplied together. When we factor a polynomial, specifically a quadratic, we're going to look for things that we can multiply together to make that quadratic. Specifically, we're going to look for things that have the form some number times x plus some other number. We're going to see how many of these we have to multiply together to make our polynomial.
Let's take a look at an example. Let's factor x2 + 4x - 12. The first part that I'm going to factor is this number that goes in front of the x. To find the number that goes in front of the x, I'm going to look at the x^2 term.
The x2 term is just 1x2, so 1 can be factored as 1 * 1. That's our only option. So each one of these 1s is going to go in front of the x in each of our two terms. To get the second number in our terms, we're going to look at the constant in our quadratic.
Factoring Quadratic Polynominals
X2 + 4x – 12
-12 + 1 = -11 ------------ -6 + 2 =-4
-1 + 12 = 11 ------------ -3 + 4 =1
-2 + 6 = 4 ------------- -4 + 3 = -1
In this factoring example, the terms must have a product of -12 and sum of 4
This -12 can be factored as -12 * 1, -1 * 12, -2 * 6, -6 * 2, -3 * 4, and -4 * 3. So anyone of these combinations could go in for the second number in each of these terms. So which one is it? To determine which one it is, we look at the middle term in our quadratic, this 4x. Now 4x has to be the sum of our two numbers. If we add these two together, 1 and -12, we get -11. 12 - 1 is 11, and so on and so forth. Now sure enough, one of these numbers is what we need, which is 4. We know that the two constants that are in our two terms are -2 and 6. So I can rewrite this as (x - 2)(x + 6). Now let's go ahead and check: (x - 2)(x + 6) is x2 + 6x - 2x - 12, which is x 2 + 4x - 12.
So the tricks to factoring quadratics: Use the a in front of your x^2 to determine the coefficients of your factored terms. You're going to use c to determine the options for your constants, and you're going to use b to determine which of those options to use. Now obviously, it can get very complicated very quickly, so you should remember that not everything can be factored, and factoring takes a lot of practice.
Lesson Review
To review, polynomials can be used to describe almost anything. The quadratics are our favorite types of functions because we can factor them, and we can solve them using the quadratic equation.