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7.3 – Simplifying Rational Expressions

Review

Reducing Fractions to Simplest Form
  • Step 1: To simplify a fraction, factor the its numerator and denominator.
  • Step 2: Look for the greatest common factor (GCF) and cancel it.
  • Ex. \frac{6}{24} can be canceled by dividing the numerator and denominator by \frac{6}{6}
    • So, \frac{6}{24}\div \frac{6}{6}=\frac{1}{4}

Notes

Rational Expressions
  • The first step in simplifying a rational expression is to factor its numerator and denominator.
  • Then cancel those factors by dividing the numerator and denominator by the GCF.
  • Example of a rational expression: \frac{x^{2}-5x-14}{x-7}

 

GCF – Greatest Common Factor
  • The GCF of a rational expression is the largest factors that can be canceled from both the numerator and denominator.
    • Ex. \frac{3x-15}{x^{2}-x-20}
    • Factor the numerator and denominator: Ex. \frac{3(x-5)}{(x-5)(x+4)}
    • Since the numerator and denominator both have the factor: (x-5), this is the GCF.

Examples

  • Ex 1. Which of the following is equal to the rational expression when x ≠ 1 or -1?
    • \frac{5(x-1)}{(x+1)(x-1)}
    • Note: The denominator cannot equal 0, which is why x ≠ 1 and x ≠ -1
    • Cancel common factors from the numerator and denominator.
    • Divide the numerator and denominator by \frac{(x-1)}{(x-1)} and they will cancel.
      • Answer: \frac{5}{x+1}
  • Ex 2. Which of the following is equal to the rational expression when x ≠ -6?
    • \frac{x^{2}-36}{x+6}
    • Note: The denominator cannot equal 0, which is why x ≠ -6
    • First, factor the numerator into: (x+6)(x-6)
    • Divide the numerator and denominator by \frac{(x+6)}{(x+6)} and they will cancel.
      • Answer: x-6
  • Ex 3. Which of the following is equal to the rational expression when x ≠ 5 or x ≠ -3?
    • \frac{x^{2}+x-6}{x^{2}-2x-15}
    • Factor the numerator and denominator and look for factors to cancel.
    • \frac{(x+3)(x-2)}{(x-5)(x+3)}
    • Cancel the (x+3) factors.
      • Answer: \frac{(x-2)}{(x-5)}
  • Ex 4. A rational expression has been simplified below.
    • \frac{(x+4)(x-2)}{9(x-2)}=\frac{x+4}{9}
    • For what values of x are the two expressions equal?
    • First, ask yourself, “What would make the denominator undefined?”
    • Second, cancel any factors that exist in the numerator and denominator.
    • The answer is “All real numbers except the number(s) that make the denominator undefined.”
    • For this problem, x-2=0 is in the denominator, so 2 would make the epxression undefined.
      • Answer: All real numbers except 2
  • Ex 5. What is the simplified form of the rational expression below?
    • \frac{6x^{2}-54}{5x^{2}+15x}
    • Factor the numerator and denominator: \frac{6(x+3)(x-3)}{5x(x+3)}
    • The GCF is (x+3), so cancel it.
      • Answer: \frac{6(x-3)}{5x}

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