- Topics At a Glance
- Exponents
- Negative Exponents
- Fractional Exponents
- Irrational Exponents
- Variables as Exponents
- Defining Polynomials
- Degrees of a Polynomial
- Multivariable Polynomials
- Degrees of Multivariable Polynomials
- Special Kinds of Polynomials
- Evaluating Polynomials
- Roots of a Polynomial
- Combining Polynomials
- Multiplying Polynomials
- Multiplication of a Monomial and a Polynomial
- Multiplication of Two Binomials
- Special Cases of Binomial Multiplication
- General Multiplication of Polynomials
- We'll Divide Polynomials Later!
**Factoring Polynomials****The Greatest Common Factor**- Recognizing Products
- Trial and Error
- Factoring by Grouping
- Summary
- Introduction to Polynomial Equations
- Solving Polynomial Equations
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

Been here and done this, but we'll go through a couple of examples to jog your memory. Sorry, we know how much your memory despises physical exercise.

Factor the polynomial *x*^{2} + *x*.

Since each term has a common factor of *x*, we can pull out the *x* and factor the polynomial as the product (*x*)(*x* + 1).

Let's make sure this is right. To check, we do the multiplication (*x*)(*x* + 1) and look to see that we have the original polynomial back:

(*x*)(*x* + 1) = *x*^{2} + *x*.

Beautiful. We now have a warm, fuzzy feeling in our chests, and hopefully it has nothing to do with that live gerbil we accidentally ingested earlier.

Factor the multivariable polynomial 6*xy* – 3*xy*^{2} + 27*x*^{2}y.

Each term has factors of 3, *x*,, and *y*. We may need a bigger pair of forceps.

Pulling out 3*xy*, we can factor the polynomial as (3*xy*)(2 – *y* + 9*x*).

Since the multiple variables make this one a little tricky, make sure we did it right. We'll multiply out the factors we found, and check that the product is the original polynomial. If it isn't, we'll figure out what we did wrong and fix it. Don't GCF (Get Completely Frantic).

(3xy)(2 – y + 9x) | = | (3xy)(2) – (3xy)(y) + (3xy)(9x) |

= | 6xy - 3xy^{2} + 27x^{2}y. |

We did it right. Not that we ever doubted ourselves.

As we go through and learn other factoring techniques, remember this one. Whenever we factor a polynomial, the first thing we want to do is look for a common factor. You can waste your time looking for *uncommon factors*, but they're rare.

Example 1

Factor the polynomial 4 |

Exercise 1

Factor 4*x*^{3} + 6*x*^{2} + 8*x*.

Exercise 2

Factor 10*x*^{5} + 25*x*^{4} + 5*x*^{3}.

Exercise 3

Factor 3*xy*^{4} + 9*x*^{2}y^{3} + 12*x*^{4}y^{5}.