Showing posts with label divide. Show all posts
Showing posts with label divide. Show all posts

Sunday, November 4, 2012

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:

Saturday, November 3, 2012

Order of Operations

When several operations are to be applied within a calculation, we must follow a specific order to ensure a single correct result.

Order of Operations:
  1. Perform all calculations within the innermost Parentheses first.
  2. Evaluate Exponent expressions.
  3. Apply the Multiplication and Division operations from left to right.
  4. Lastly, work all Addition and Subtraction operations from left to right.
Caution:  Please do not dismiss the fact that multiplication and division should be worked from left to right.  Many standardized exams will test you on this fact.  The following example illustrates the problem.

Instructional Video: Order of Operations - The Basics

Order of operation problems get a bit more tedious when fractions are involved. Remember that when adding or subtracting fractions you need to first find the equivalent fractions with a common denominator. Multiplication does not require a common denominator.


We will see that some of the problems have different looking parentheses { [ ( ) ] }, treat them the same and just remember to perform the innermost parentheses first.  Some problems may involve an absolute value, in which case you will need to apply the innermost absolute value first as you would if it were a parentheses.
To avoid these unnecessary mistakes, work one operation at a time and for each step rewrite everything.  This may seem like a lot of work but it really helps avoid errors.

Video Examples on YouTube: