Up

Simplifying and excluded values

One of the first things we would like to do with rational expressions is learn to simplify them.  Rational expressions can look rather intimidating, but with a little bit of simplification, we can make them look a whole lot less intimidating.

                We do have to watch out for excluded values when we are dealing with rational expressions.  Excluded values are those values of the variable which would cause the denominator to be 0.  This is not allowed, since division by 0 is undefined.  So as we work to simplify these rational expressions, we must be careful to state the excluded values.

                Let’s take a look at an example.  Simplify the expression

\(\Large \frac{{{r^2} - 15r + 50}}{{{r^2} - 3r - 70}}\)

The first thing we should do is factor the polynomials in both the numerator and the denominator.  If we are lucky, we will find one or more common factors that can cancel.  We have

\(\Large \frac{{{r^2} - 15r + 50}}{{{r^2} - 3r - 70}} = \Large \frac{{\left( {r - 10} \right)\left( {r - 5} \right)}}{{\left( {r - 10} \right)\left( {r + 7} \right)}}\)

The polynomials do factor!  Okay, before we go any further we want to state the excluded values of this rational expression.  Again, excluded values are those values that would cause the denominator to be 0.  In our case, the excluded values would be those values of \(r\) such that \({r^2} - 3r - 70\) or \(\left( {r - 10} \right)\left( {r + 7} \right)\) is 0. We set \(\left( {r - 10} \right)\left( {r + 7} \right) = 0\) and we have the excluded values are -7 and 10. Now let’s finish simplifying this rational expression by cancelling the common factor \(\left( {r - 10} \right)\). We have

\(\Large \frac{{{r^2} - 15r + 50}}{{{r^2} - 3r - 70}} = \Large \frac{{\left( {r - 10} \right)\left( {r - 5} \right)}}{{\left( {r - 10} \right)\left( {r + 7} \right)}}\),                 with \(r \ne  - 7,10\)

\( = \Large \frac{{r - 5}}{{r + 7}}\),                   with \(r \ne  - 7,10\)

We have just simplified a rational expression!  Let’s try another example.  We want to simplify the expression

\(\Large \frac{{{m^2} - 4m - 5}}{{5m + 5}}\)

Let’s begin again by doing any factoring that we can do.  We have

\(\Large \frac{{{m^2} - 4m - 5}}{{5m + 5}} = \Large \frac{{\left( {m - 5} \right)\left( {m + 1} \right)}}{{5\left( {m + 1} \right)}}\)

Now we see that there is a common factor of \(\left( {m + 1} \right)\) that we can cancel.  We also should be able to see that the excluded value is -1 (since \(m =  - 1\) would result in the denominator being 0). We finish the problem:

\(\Large \frac{{{m^2} - 4m - 5}}{{5m + 5}} = \Large \frac{{\left( {m - 5} \right)\left( {m + 1} \right)}}{{5\left( {m + 1} \right)}}\),            with \(m \ne  - 1\)

\( = \Large \frac{{m - 5}}{5}\),       with \(m \ne  - 1\)

Below you can download some free math worksheets and practice.


Downloads:
1655 x

Simplify each and state the excluded values.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Rational-Expressions-Simplifying-and-excluded-values-easy


Watch below how to solve this example:

 

Downloads:
1414 x

Simplify each and state the excluded values.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Rational-Expressions-Simplifying-and-excluded-values-medium

Watch below how to solve this example:

 

Downloads:
1273 x

Simplify each and state the excluded values.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Rational-Expressions-Simplifying-and-excluded-values-hard

Watch below how to solve this example:

 
 
 

Facebook PageGoogle PlusTwitterYouTube Channel

Algebra and Pre-Algebra

Beginning Algebra
Adding and subtracting integer numbers
Dividing integer numbers
Multiplying integer numbers
Sets of numbers
Order of operations
The Distributive Property
Verbal expressions
Beginning Trigonometry
Finding angles
Finding missing sides of triangles
Finding sine, cosine, tangent
Equations
Absolute value equations
Distance, rate, time word problems
Mixture word problems
Work word problems
One step equations
Multi step equations
Exponents
Graphing exponential functions
Operations and scientific notation
Properties of exponents
Writing scientific notation
Factoring
By grouping
Common factor only
Special cases
Linear Equations and Inequalities
Plotting points
Slope
Graphing absolute value equations
Percents
Percent of change
Markup, discount, and tax
Polynomials
Adding and subtracting
Dividing
Multiplying
Naming
Quadratic Functions
Completing the square by finding the constant
Graphing
Solving equations by completing the square
Solving equations by factoring
Solving equations by taking square roots
Solving equations with The Quadratic Formula
Understanding the discriminant
Inequalities
Absolute value inequalities
Graphing Single Variable Inequalities
Radical Expressions
Adding and subtracting
Dividing
Equations
Multiplying
Simplifying single radicals
The Distance Formula
The Midpoint Formula
Rational Expressions
Adding and subtracting
Equations
Multiplying and dividing
Simplifying and excluded values
Systems of Equations and Inequalities
Graphing systems of inequalities
Solving by elimination
Solving by graphing
Solving by substitution
Word problems