Rational Expression

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

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

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

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 ${\frac {x^{2}-1}{x+1}}$ . We may simplify this in the following way:

${\frac {x^{2}-1}{x+1}}={\frac {(x+1)(x-1)}{x+1}}=x-1,\qquad x\neq -1$

This is nice because we have obtained something we understand quite well, $x-1$ , from something we didn't.

Resources

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