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 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{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

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 p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0}

and

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 q(x)=b_mx^m+b_{m-1}x^{m-1}+\cdots+b_1x+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

Rational Expression