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 ${\frac {4x^{3}}{12x^{2}}}$ 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

$p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}$

and

$q(x)=b_{m}x^{m}+b_{m-1}x^{m-1}+\cdots +b_{1}x+b_{0}$

When we take the quotient of the two we obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}}

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{x^2-1}{x+1}} . We may simplify this in the following way:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{x^2-1}{x+1}=\frac{(x+1)(x-1)}{x+1}=x-1,\qquad x\ne -1}

This is nice because we have obtained something we understand quite well, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x-1} , from something we didn't.

Resources

• Algebra, Wikibooks: Calculus
• Identify and Simplify Rational Expressions, Lumen Learning
• Simplifying Rational Expressions, The Organic Chemistry Tutor