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College Algebra: Help and Review27 chapters | 228 lessons

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Lesson Transcript

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.

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 2*x* + 4, *x*^2 + 3*x* + 2, and *x*^5 + 2*x*^2.

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 + 3*x* + 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 + 3*x* + 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.

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.

A third method you can use is the grouping method if your polynomial has four terms. We can use this method to factor a polynomial, such as *x*^3 + 2*x*^2 + 2*x* + 4. To use this method, we group our four terms into groups of two like this; (*x*^3 + 2*x*^2) + (2*x* + 4). We then use the greatest common factor method to factor each set of parentheses.

The greatest common factor in the first set of parentheses is *x*^2, and the greatest common factor in the second is 2. We rewrite our polynomial as *x*^2(*x* + 2) + 2(*x* + 2). Notice how we now have two compound terms, *x*^2(*x* + 2) and 2(*x* + 2), where each has a common factor of (*x* + 2). We can now repeat the greatest common factor method and factor out the (*x* + 2). We get (*x* + 2) (*x*^2 + 2). And we are done.

If we see a polynomial in one of the forms above, and we can't use the particular method mentioned above, then that means that the polynomial can't be factored. If we liken factoring a polynomial to stretching a muscle, then that means there are no exercises to strengthen that particular polynomial. The muscles of that polynomial are way too tight to stretch.

What have we learned? We've learned that **polynomial expressions** are expressions made of terms that are the product of variables and a number. Just like there are different exercises to stretch different body parts, so there are different methods of factoring for different kinds of polynomials.

Here is a summary of the different methods we learned in this video lesson:

Method | Example Polynomial | Description |
---|---|---|

Quadratic | x^2 + 3x + 2 |
You look for two numbers that multiply to give you the last number and add to the middle number. This polynomial factors into (x + 1)(x + 2). |

Greatest Common Factor Method | x^5 + x^3 |
Look for a common factor that you can divide each term by. This polynomial factors into x^3(x^2 + 1). |

Grouping Method | x^3 + 2x^2 + 2x + 4 |
You group into pairs of two terms and then use the greatest common factor method for each grouping, and then repeat the greatest common factor method for the two remaining compound terms. |

After watching this lesson, you should be able to apply the greatest common factor, grouping and quadratic methods to factor polynomials.

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College Algebra: Help and Review27 chapters | 228 lessons

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