Factoring Polynomial Expressions

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  • 0:01 Polynomial Expressions
  • 0:50 Factoring Quadratics
  • 2:50 Greatest Common Factor Method
  • 3:35 Grouping Method
  • 5:05 Lesson Summary
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Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

When it comes to factoring polynomials, there are several methods to choose from depending on what kind of polynomial you are looking at. Watch this video lesson to learn how to identify and use the appropriate method.

Polynomial Expressions

In algebra, being able to factor polynomial expressions, expressions made of terms that are the product of variables and a number, is an essential skill. It's like us. To grow up, we need to learn how to walk. In algebra, to grow in our skills, we need to know how to factor. Factoring is about breaking up our polynomial into its parts. Just like we are made up of our body parts, such as our arms and legs, so our polynomials are made up of its factors.

Also, just like there are different exercises we can do to stretch each muscle, so we have different factoring methods depending on the kind of polynomial we are looking at. Examples of polynomials include 2x + 4, x^2 + 3x + 2, and x^5 + 2x^2.

Factoring Quadratics

The first method we will look at is the quadratic method for factoring polynomials that are quadratics and whose first coefficient is 1. These are the polynomials whose highest degree, or exponent, is a 2. So, a polynomial, such as x^2 + 3x + 2, is an example of a quadratic whose first coefficient is 1.

Notice the highest exponent of 2 that we have and the coefficient of 1 for the x^2 term. Quadratic polynomials are made up of two factors, and by using this quadratic method, we can find them if they are there. Not all quadratic polynomials factor nicely into their two parts. If they do, then this method will find what they are.

To use this method, you first identify your three numbers, your three coefficients. Because quadratics have a general algebraic form of ax^2 + bx + c, we are looking for the values of a, b, and c. Our first value, a, is 1 and we look for two numbers that multiply to the c value and add to the b value. In our example, our a is 1, our b is 3, and our c is 2. So we ask what two numbers multiply together to make 2 and add together to get 3? Isn't it 1 and 2? 1 times 2 is 2, and 1 plus 2 is 3! So x^2 + 3x + 2 factors into (x + 2)(x + 1).

Do you see how we have written our numbers 1 and 2? That is how you factor a quadratic polynomial where the first coefficient number, a, is 1. Now remember, your quadratic polynomial does not necessarily need all three terms. It could have just two terms and you may still be able to use this method. If you can't find two numbers that fit, then you may have to use another method.

Greatest Common Factor Method

The second method we will look at is the greatest common factor method. This works for any kind of polynomial if all the terms in the polynomial have a common factor. For example, we can use this method for the polynomial x^5 + x^3 because all the terms of this polynomial have the common factor of x^3. I can divide each term by x^3. If I do that, x^5 + x^3 factors into x^3(x^2 + 1). Notice how I've written what I've divided each term by on the outside of the parentheses, and I wrote what I get after dividing each term inside the parentheses. This is how you factor using this method.

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