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Simplify Rational Expressions

This guide provides an overview of simplifying rational expressions, defined as fractions with polynomial numerators and denominators. Key concepts include finding the domain of rational expressions, which excludes values making the denominator zero. The process of simplification involves dividing out common factors in the numerator and denominator. Examples illustrate how to factor polynomials and determine their reduced forms. Remember, you can only cancel out factors, not sums or differences. Master these techniques for effective simplification of rational expressions.

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Simplify Rational Expressions

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  1. 11.5 Simplify Rational Expressions

  2. Rational Expression: a fraction whose numerators and denominators are polynomials. Examples: 3 , 2x , and 6 . x + 4 x2 – 9 x2 + 1

  3. Domain: the set of all real numbers except those for which the denominator is zero. For example: The domain of 3 is x + 4 all real numbers except -4.

  4. Simplify Rational Expression the same way that you reduce a fraction. Divide the numerator and denominator by “x” 25 = Divide the numerator and denominator by “2” 2x + 4 =

  5. Domain:x can be any real number except a number that would make the denominator 0! Find the domain of each expression below.

  6. How do you know when your rational expression is reduced? Reduced Form: when the numerator and denominators have no common factors except for ±1.

  7. The RULE: To simplify fractions, divide out common factors. REMEMBER you can’t cancel out +(adding) and –(subtracting)

  8. For a,b,c any non-zero real numbers, Remember you can only cancel out FACTORS!!!

  9. Simplify the expression. x2 + 4x +4 Factor each polynomial. x2 - 4 (x + 2)(x + 2)(x + 2)(x – 2) What does the numerator and denominator have in common? Factor out (x + 2) The answer is: x + 2 x - 2

  10. Simplify the expression. Factor each polynomial. x2 - 7x + 12x2 +3x - 18 (x - 4)(x – 3)(x + 6)(x - 3) What does the numerator and denominator have in common? Factor out (x – 3) The answer is: x - 4 x + 6

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