Showing posts with label polynomial. Show all posts
Showing posts with label polynomial. Show all posts

Introduction to Factoring and Factor by Grouping

Factoring is one of the more important skills to learn in a beginning algebra course. The idea is to work the distributive property in reverse. The goal is to write polynomials as products of simpler polynomials.
To decide what the GCF (Greatest Common Factor) is, look for the largest factor that divides into all the terms. In other words, what common term will divide into all the given terms evenly?
In the above example, the variable z is not common to all the terms so it is not included in the GCF. Use the smallest exponent for the common variables x and y.

Identify the GCF.
Factor out the GCF:

   Step 1: Identify the GCF.
   Step 2: Divide each term by the GCF to determine the remaining factor.
   Step 3: Check by multiplying.
The check is not really necessary; however, it is good practice to at least check the factoring mentally.

Factor the GCF out of the expression.
Tip: Be careful to use a 1 when the entire term factors out. This is where a mental check is important. Be sure that if you were to distribute you would get back to the original expression.
All of the above problems require only one step. The hard part is to identify the GCF. Here is the check for the last solved problem.
Factoring 4-Term Polynomials by Grouping
Now we will use the idea of factoring out the GCF in a technique called factoring by grouping of four-term polynomials. The steps are as follows.

   Step 1: Group the first two terms and the last two terms.  Factor out the GCF of both groupings.
   Step 2: If the remaining binomial factors are the same factor it out.
   Step 3: Check by multiplying.
Factor by grouping.
Sometimes you will encounter 4-term polynomials where factoring by grouping does not seem to work.  When this is the case, you can try to rearrange the terms in a different order and try again. For example, try to factor the following polynomial  by grouping.

Rearrange the terms and try again.
You can check by multiplying.
Video Examples on YouTube:

Dividing Polynomials

In this section we will learn how to divide polynomials. Students find this to be one of the more difficult topics in Algebra. Plan on spending some extra time reviewing the techniques and solutions presented here. (Assume all expressions that appear in a denominator are nonzero.)

Dividing by a Monomial
When dividing we will be using the quotient rule,
This says that when dividing two expressions with the same base you subtract exponents.
In fact, we are using the property for adding fractions with a common denominator
By breaking up the fraction we could then simply and then cancel.

A common mistake would be to cancel denominator with only one of the terms. We are dividing the entire expression in the numerator so every term must be cancelled with the denominator.

Dividing by a Polynomial
In this section, we will use polynomial long division when dividing by something other than a monomial. The good news is that the steps are basically the same as the regular division algorithm we are already used to. Use polynomial long division to divide the following.

The completed process follows.
Division does not always work out so evenly, sometimes you will have a remainder.
Some of the polynomials will be missing terms. In other words, not all the exponents will be there. When first learning, it really is best to use placeholders and include the missing terms using zero as coefficients.

Polynomial long division takes some practice to master. Be patient, do lots of problems and soon you will find them to be enjoyable.

Click here for an introduction to synthetic division.

Video Examples on YouTube: