Rational Expressions

Understanding Rational Expressions

  • A Rational Expression is any expression or equation that can be written as a ratio of two polynomials.
  • The expression in the numerator and the denominator may be a monomial, binomial or a higher degree polynomial.
  • Like a rational number, which has all real numbers as its domain except for zero, a rational expression has all real numbers as its domain except for any value that makes the denominator equal to zero.
  • Rational expressions are used in a wide array of problems in algebra, from simple equations to complex real-world word problems.

Simplifying a Rational Expression

  • To simplify a rational expression, we factorise the numerator and the denominator and cancel out common factors.
  • Be cautious, only multiplication allows the cancellation of terms. Cancellation is mistaken when it is tried in addition or subtraction.

Adding and Subtracting Rational Expressions

  • Before adding or subtracting rational expressions, we must first find a common denominator.
  • The common denominator is the least common multiple (LCM) of the two denominators.
  • Once the common denominator is obtained, rewrite each term as an equivalent fraction with the common denominator. Then add or subtract the numerators of these fractions.

Multiplying and Dividing Rational Expressions

  • When multiplying rational expressions, we simply multiply the numerators together and the denominators together, and then simplify the result, if possible.
  • Rational expressions are divided by multiplying the first expression by the reciprocal of the second.
  • As with multiplication, we simplify if possible by cancelling common factors in the numerator and the denominator.

Evaluating Rational Expressions

  • Provide the value of the variable, substitute this value into the expression and simplify to get the result.
  • Ensure the value of the variable is not one which makes the denominator zero - keep in mind the domain of rational expressions.

Solving Rational Expressions

  • When solving rational equations, multiply every term by the common denominator to eliminate the denominators.
  • Remember to check your answers, as this method can introduce extraneous solutions - solutions that do not satisfy the original equation.

Graphing Rational Expressions

  • When graphing rational functions, it is important to identify any vertical asymptotes and horizontal asymptotes, as well as any potential holes in the graph.
  • These are identified by factors that cancel out in your simplified rational expression. A factor that cancels out will be a hole, while a factor that does not cancel will be an asymptote.
  • Plot a few points to help complete the sketch, especially at points close to the asymptotes or the hole.