Throughout Algebra 1 and 2, one of the key concepts you?ll learn is how to factor polynomials including but not limited to quadratics. Factoring is an important process that helps us understand more about our equations. Through factoring, we rewrite our polynomials in a simpler form, and when we apply the principles of factoring to equations, we yield a lot of useful information.
For example from the factored form, you can easily identify solutions. You get an idea of the quantity and type of solutions an equation has. You can find its roots/x-intercepts, which makes it easier to graph. You can even tell a bit about the behavior of the graph through multiplicity. All of these are wonderful concepts, but before we get to them it?s important to master the art of factoring.?
There are a lot of different factoring techniques. From the standard here?s how to factor a quadratic to the less obvious techniques of completing the square and polynomial long division, a lot of what you learn in algebra revolves around manipulating equations, so that?s what we are focusing on today!
The Basics: How to Factor Quadratics
In this tutorial, I show you the very basics of factoring a quadratic. You will learn how to choose your factors and check your solution.
It?s important to note that in these examples, I?m not working with the full equation (notice there are no equal signs). This is a common first step as you are learning how to factor. We can set any of these polynomials equal to zero and then we have an equation. With that equation, we can do the same factoring process, and take it a step further to find the solution (if it exists) by solving for the variable, and then we can graph the quadratic if we wish to do so.
NOTE: For factoring quadratics where there is a coefficient on the x-squared term, check out the ?acb method? under tricky quadratics towards the end of this post!
The Basics: How to Factor out the Greatest Common Factor (GCF)
Factoring out the greatest common factor is a handy little technique that you can use whenever there is a factor (number, variable, or both) common to ALL terms in your polynomial.
The interesting thing about this technique is that as you move on to more difficult polynomials and equations, you?ll find that often this step can be used in conjunction with other types of factoring to get your polynomial down to the very simplest form.
Formulas: How to Factor the Difference of Perfect Squares
Some types of factoring fall nice and neatly into a handful of formulas. These formulas can act as guides for factoring certain special types of polynomials. Unfortunately, it might take a bit of memorization to get these ones down.
In the video tutorial, you?ll learn about a few different formulas. These include:
- Difference of Perfect Squares Formula
- Perfect Square Trinomials Formula (positive & negative versions)
Cool Tricks: How to Factor by Grouping
Factor by grouping is a cool technique that you can use in a couple of interesting ways. You can use it to factor down polynomials with four terms, like the examples in the video, by first factoring out a GCF from two pairs of terms.
If after factoring out the GCFs you are left with two identical binomials in parenthesis, then that means your polynomial can be factored by grouping, and you can go ahead and finish the factorization.
The other interesting way you can use factor by grouping is by taking a normal trinomial quadratic and splitting the middle x-term into two terms so that they still sum to the original middle term. This will give you 4 terms and the opportunity to apply factor by grouping. I don?t cover this approach in the tutorial because it isn?t the traditional use of factor by grouping, but I do know that some people really love solving quadratics that way.
Tricky Quadratics: How to Factor Quadratics with Leading Coefficient greater than 1 | The ACB Method
In this tutorial, we take a look at one of the more difficult types of quadratics to factor: when we stumble upon a quadratic with a value on the x-squared term.
What you?ll notice when you try to factor these types of polynomials is that the extra coefficient creates a lot more possibilities of both factors and where to place those factors. Now you can still factor these types of problems using a bit of guess and check alongside the traditional quadratic factoring method, but you?ll notice that it can get pretty tricky.
Fortunately, we have a tried and true method for tackling these types of problems called the ACB Method (or CAB method). This method is the secret to factoring these complicated quadratics. Using this technique takes away all the guesswork, and gives you a step-by-step process you can follow every time you encounter these types of quadratics ?
Advanced Techniques: Complete the Square
This is not technically a factoring technique, but I do think it is so closely related to factoring that I wanted to include it in this guide.
Completing the square is a method you can use to rewrite a quadratic from standard form (i.e. y=ax+bx+c) to vertex form (i.e. y=(x-h) + k) so technically you are not rewriting your equation in a fully factored form since you?ll have that constant (k) hanging out on the outside of your binomial squared. BUT you are converting your equation into a form where you can easily solve for the x-intercepts and graph, so it?s very similar to the desired goal for factoring.
If you want to explore this topic further and see it in action, check out the tutorial below!
Advanced Techniques: Polynomial Long Division
This is another technique that isn?t explicitly factoring, but it can be used to rewrite a polynomial in factored form. Polynomial long division is the process used to divide a polynomial by a smaller polynomial, most typically a binomial.
In this type of Algebra 2 problem, one could use the Rational Zero Theorem to identify potential solutions, and then use a process such as polynomial long division to test and divide out the solutions. Typically this process is done multiple times until you have identified all of the solutions.
Now if you think about dividing a value (the dividend) by another value (the divisor), the resulting value (the quotient) is really a factor of the dividend along with the divisor.
A and C become the factorization of B when rewritten
So you can use division to factor a polynomial. To learn how to perform polynomial long division, check out this tutorial.
Whew, that?s a lot!
Wow, we covered a lot of material today! If you are new to factoring, I suggest focusing on one topic at a time and make sure you get lots of practice with different problems before moving on to the next topic.
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