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Chapter 8 Rational Expressions

Chapter 8 Rational Expressions. § 8.1. Rational Expressions and Their Simplification. Rational Expressions. A rational expression is the quotient of two polynomials. Some examples of rational expressions are:. Blitzer, Introductory Algebra , 5e – Slide # 3 Section 8.1.

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Chapter 8 Rational Expressions

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  1. Chapter 8Rational Expressions

  2. §8.1 Rational Expressions and Their Simplification

  3. Rational Expressions A rational expression is the quotient of two polynomials. Some examples of rational expressions are: Blitzer, Introductory Algebra, 5e – Slide #3 Section 8.1

  4. Excluding Values from Rational Expressions If a variable in a rational expression is replaced by a number that causes the denominator to 0, that number is excluded as a replacement for the variable. The rational expression is undefined at any value that produces a denominator of 0. Danger: Beware of 0s in Denominators Blitzer, Introductory Algebra, 5e – Slide #4 Section 8.1

  5. Excluding Numbers from Rational Expressions Remember…A rational expression is undefined for all values that make the denominator 0. Set the denominator equal to 0. Add 5 to both sides. Divide both sides by 2. Blitzer, Introductory Algebra, 5e – Slide #5 Section 8.1

  6. Excluding Numbers from Rational Expressions EXAMPLE Set the denominator equal to 0. Factor. Set each factor equal to 0 and solve. Blitzer, Introductory Algebra, 5e – Slide #6 Section 8.1

  7. Rational Expressions - Domain EXAMPLE Find the domain of f if SOLUTION The domain of f is the set of all real numbers except those for which the denominator is zero. We can identify such numbers by setting the denominator equal to zero and solving for x. Set the denominator equal to 0. Factor. Blitzer, Introductory Algebra, 5e – Slide #7 Section 8.1

  8. Rational Expressions - Domain CONTINUED or Set each factor equal to 0. Solve the resulting equations. Because 4 and 9 make the denominator zero, these are the values to exclude. Thus, Blitzer, Introductory Algebra, 5e – Slide #8 Section 8.1

  9. Rational Expressions Blitzer, Introductory Algebra, 5e – Slide #9 Section 8.1

  10. Simplifying Rational Expressions • Factor the numerator and denominator. • Divide both the numerator and denominator by any common factors. Blitzer, Introductory Algebra, 5e – Slide #10 Section 8.1

  11. Simplifying Rational Expressions EXAMPLE Factor the numerator and denominator. Divide both the numerator and denominator by the common factor, x-4. Blitzer, Introductory Algebra, 5e – Slide #11 Section 8.1

  12. Factors that are Opposites in Sign Simplifying Rational Expressions with Opposite Factors in the Numerator and Denominator The quotient of two polynomials that have opposite signs and are additive inverses is -1. You may cancel factors that are negatives of each other, but remember that you get a quotient of -1 when you do this. Blitzer, Introductory Algebra, 5e – Slide #12 Section 8.1

  13. Factors that are Opposites Simplify a Rational Expression EXAMPLE The numerator and denominator are opposites, or additive inverses. They differ only in their signs. Factor the numerator and denominator. Divide both the numerator and denominator by the common factor, 2-x. Blitzer, Introductory Algebra, 5e – Slide #13 Section 8.1

  14. Simplifying Rational Expressions EXAMPLE Simplify: SOLUTION Factor the numerator and denominator. Divide out the common factor, x + 1. Simplify. Blitzer, Introductory Algebra, 5e – Slide #14 Section 8.1

  15. Simplifying Rational Expressions EXAMPLE Simplify: SOLUTION Factor the numerator and denominator. Rewrite 3 – x as (-1)(-3 + x). Rewrite -3 + x as x – 3. Blitzer, Introductory Algebra, 5e – Slide #15 Section 8.1

  16. Simplifying Rational Expressions CONTINUED Divide out the common factor, x – 3. Simplify. Blitzer, Introductory Algebra, 5e – Slide #16 Section 8.1

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