Difference between revisions of "Rational Expression"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
 
(5 intermediate revisions by one other user not shown)
Line 1: Line 1:
 +
See also: [[Rational Equations]], [[Rational Functions]], [[Graphs of Rational Functions]]
 +
 
==Rational Expression==
 
==Rational Expression==
 
Rational expressions are fractions that have a polynomial in the numerator, denominator, or both. Although rational expressions can seem complicated because they contain variables, they can be simplified using the techniques used to simplify expressions such as <math>\frac{4x^3}{12x^2}</math> combined with techniques for factoring polynomials.
 
Rational expressions are fractions that have a polynomial in the numerator, denominator, or both. Although rational expressions can seem complicated because they contain variables, they can be simplified using the techniques used to simplify expressions such as <math>\frac{4x^3}{12x^2}</math> combined with techniques for factoring polynomials.
Line 7: Line 9:
 
* Factor the numerator and denominator.
 
* Factor the numerator and denominator.
 
* Find common factors for the numerator and denominator and simplify.
 
* Find common factors for the numerator and denominator and simplify.
 +
 +
Consider two polynomials
 +
<center><math>p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0</math></center> and
 +
<center><math>q(x)=b_mx^m+b_{m-1}x^{m-1}+\cdots+b_1x+b_0</math></center>
 +
When we take the quotient of the two we obtain
 +
<center><math>\frac{p(x)}{q(x)}=\frac{a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0}{b_mx^m+b_{m-1}x^{m-1}+\cdots+b_1x+b_0}</math></center>
 +
The ratio of two polynomials is called a '''rational expression'''. Many times we would like to simplify such a beast. For example, say we are given <math>\frac{x^2-1}{x+1}</math> . We may simplify this in the following way:
 +
<center><math>\frac{x^2-1}{x+1}=\frac{(x+1)(x-1)}{x+1}=x-1,\qquad x\ne -1</math></center>
 +
This is nice because we have obtained something we understand quite well, <math>x-1</math> , from something we didn't.
  
 
==Resources==
 
==Resources==
 
* [https://courses.lumenlearning.com/beginalgebra/chapter/6-1-1-introduction-to-rational-expressions/ Identify and Simplify Rational Expressions], Lumen Learning
 
* [https://courses.lumenlearning.com/beginalgebra/chapter/6-1-1-introduction-to-rational-expressions/ Identify and Simplify Rational Expressions], Lumen Learning
 
* [https://www.youtube.com/watch?v=uVpsz-xpnPo Simplifying Rational Expressions], The Organic Chemistry Tutor
 
* [https://www.youtube.com/watch?v=uVpsz-xpnPo Simplifying Rational Expressions], The Organic Chemistry Tutor
 +
 +
== Licensing ==
 +
Content obtained and/or adapted from:
 +
* [https://en.wikibooks.org/wiki/Calculus/Algebra Algebra, Wikibooks: Calculus] under a CC BY-SA license

Latest revision as of 22:44, 13 November 2021

See also: Rational Equations, Rational Functions, Graphs of Rational Functions

Rational Expression

Rational expressions are fractions that have a polynomial in the numerator, denominator, or both. Although rational expressions can seem complicated because they contain variables, they can be simplified using the techniques used to simplify expressions such as combined with techniques for factoring polynomials.

Simplifying Rational Expression

To simplify a rational expression, follow these steps:

  • Determine the domain. The excluded values are those values for the variable that result in the expression having a denominator of 0.
  • Factor the numerator and denominator.
  • Find common factors for the numerator and denominator and simplify.

Consider two polynomials

and

When we take the quotient of the two we obtain

The ratio of two polynomials is called a rational expression. Many times we would like to simplify such a beast. For example, say we are given . We may simplify this in the following way:

This is nice because we have obtained something we understand quite well, , from something we didn't.

Resources

Licensing

Content obtained and/or adapted from: