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Wize High School Algebra I Textbook (Common Core) > Rational Expressions and Functions

Rational Expressions and Functions

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Multiplying and Dividing Rational ExpressionsNext Section
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Rationa Expressions & Functions

A rational expression is a quotient (or ratio) of two polynomials, where the deonominator is not 0.

Examples
  • 2x+1x−5,  x≠5\dfrac{2x+1}{x-5},~~x\neq5
  • 3x2+2x−15x3−5,  x≠1\dfrac{3x^2+2x-1}{5x^3-5},~~x\neq1
  • 1x2+3\dfrac{1}{x^2+3}


A rational function is any function that is a ratio of two polynomials, where the denominator is not 0.
f(x)=P(x)Q(x),  Q(x)≠0f(x)=\dfrac{P(x)}{Q(x)},~~Q(x)\neq 0
Examples
  • f(x)=2x+1x−5,  x≠5f(x)=\dfrac{2x+1}{x-5},~~x\neq5
  • g(x)=3x2+2x−15x3−5,  x≠1g(x)=\dfrac{3x^2+2x-1}{5x^3-5},~~x\neq1
  • h(x)=1x2+3h(x)=\dfrac{1}{x^2+3}

Simplifying Rational Expressions & Functions

We can simplify rational expressions and rational functions by dividing out any common factors that belong to both numerator and denominator.

Example
Simplify x2+3x+2x2−1\dfrac{x^2+3x+2}{x^2-1}

First, we factor fully:

=(x+1)(x+2)(x+1)(x−1)=\dfrac{(x+1)(x+2)}{(x+1)(x-1)}

Then, we divide out any common factor:

=(x+1)(x+2)(x+1)(x−1)=\dfrac{\cancel{(x+1)}(x+2)}{\cancel{(x+1)}(x-1)}

=x+2x−1=\dfrac{x+2}{x-1}

The restrictions are that

Practice: Evaluating Rational Functions

Consider f(x)=x2−1x−1f(x)=\dfrac{x^2-1}{x-1}.
a) evaluate f(2)f(2)
b) evaluate f(−2)f(-2)
c) state any restrictions on the variable xx.
Evaluate f(2)f(2).

Practice: Simplifying Rational Expressions

Consider the rational expression 4x2−12x+92x2−x−3\dfrac{4x^2-12x+9}{2x^2-x-3}.
Simplfy this expression fully.

Practice: Simplifying Rational Expressions

Consider the rational expression x3−3x2y−10xy2x2−25y2\dfrac{x^3-3x^2y-10xy^2}{x^2-25y^2}.
Simplify the expression fully.
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Practice: Simplifying Rational Expressions

Is there a rational expression that simplifies to x+1x−2\dfrac{x+1}{x-2}, with denominator of the form ax2+bx+cax^2+bx+c, and has the restriction

a) x≠−1x\neq -1 and x≠2x\neq 2 only?

b) x≠2x\neq 2 only?