RatF - Rational Expressions and Operations Lesson

Rational Expressions and Operations

Rational Expressions

A rational expression is a ratio of two polynomial expressions. Simply put we can have a fraction where the numerator and denominator are both polynomial expressions. From our first exposure to fractions we found that we can reduce them. This carries over to algebraic fractions, where we are able to simplify them. Just like with number fractions, algebraic fractions are reduced by "canceling" factors common to both the numerator and the denominator.

Remember that polynomials are factors, and therefore, one polynomial factor can be canceled with the exact same polynomial factor.

For example, equation image indicator $LaTeX: \frac{\left(x-1\right)}{\left(x-1\right)}=1$ $\frac{\left(x-1\right)}{\left(x-1\right)}=1$ while $LaTeX: \frac{\left(x+1\right)}{\left(x-1\right)}\ne1$ $\frac{\left(x+1\right)}{\left(x-1\right)}\ne1$ . In the first rational expression, the polynomial in the numerator is exactly equal to the polynomial denominator. But, in the second rational expression the polynomial in the numerator is not exactly equal to the polynomial denominator.

Here are some more examples of rational expressions: equation image indicator $LaTeX: \frac{6}{x-1},\frac{z^2-1}{z^2+5},\frac{4x^2+6x-10}{1},\frac{m^4+18m+1}{m^2-m-6}$ $\frac{6}{x-1},\frac{z^2-1}{z^2+5},\frac{4x^2+6x-10}{1},\frac{m^4+18m+1}{m^2-m-6}$

We must also be aware when working with fractions, the "excluded values" //cms.gavirtualschool.org/DEV15/Math/CCGPSAlgebra2/Softchalks/03_RationalandRadicalRelationships/ada-reference.gif must be considered. Excluded values are numbers that would make the denominator zero, and are usually written in form " ". The denominator of a fraction can never be zero (try something divided by 0 in your calculator). Excluded values are also called restrictions, because the function is not a function at the certain values for x that result in a "0" in denominator. Excluded values //cms.gavirtualschool.org/DEV15/Math/CCGPSAlgebra2/Softchalks/03_RationalandRadicalRelationships/ada-reference.gif determine the domainof the function.

Here are some examples of rational expression being simplified or already simplified along with restrictions:

$LaTeX: \frac{3t}{7y},\:y\ne0$ $\frac{3t}{7y},\:y\ne0$
$LaTeX: \frac{\left(m+11\right)}{\left(m+7\right)},\:m\ne-7$ $\frac{\left(m+11\right)}{\left(m+7\right)},\:m\ne-7$
$LaTeX: \frac{3a+4}{4a^2-25}=\frac{3a+4}{\left(2a+5\right)\left(2a-5\right)}=a\ne-\frac{5}{2},\:\frac{5}{2}$ $\frac{3a+4}{4a^2-25}=\frac{3a+4}{\left(2a+5\right)\left(2a-5\right)}=a\ne-\frac{5}{2},\:\frac{5}{2}$

Each of the excluded values would make the denominators zero. When reducing fractions, even if a factor in the denominator cancels, it must be considered for an excluded value because the original fraction would be undefined for that number.

Example

equation image indicator $LaTeX: \frac{y^2-25}{y^2-3y-40}=\frac{\left(y+5\right)\left(y-5\right)}{\left(y-8\right)\left(y+5\right)}=\frac{\left(y-5\right)}{\left(y-8\right)}=y\ne-5,\:8$ $\frac{y^2-25}{y^2-3y-40}=\frac{\left(y+5\right)\left(y-5\right)}{\left(y-8\right)\left(y+5\right)}=\frac{\left(y-5\right)}{\left(y-8\right)}=y\ne-5,\:8$

Excluded values can also be written as a domain. The form often used is something like this $LaTeX: \left\{x|x\ne-5,\:8\right\}$ $\left\{x|x\ne-5,\:8\right\}$ , which is read "for all x such that x is not equal to -5 and 8".

The information just discussed will help you greatly in simplifying rational expressions.

Now, to simplify a rational expression:

Factor everything that can be factored.
Simplify the monomials (use exponent rules).
Eliminate common binomial factors.

The following videos will discuss and model in greater detail the concepts surrounding the simplification of rational expressions. After watching these teaching videos, practice items are provided for you to complete to master these concepts.

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