Rational Expressions and Equations Chapter 6

§ 6.1 Simplifying, Multiplying, and Dividing

Tobey & Slater, Intermediate Algebra, 5e - Slide #3 Rational Expressions or Fractional algebraic expression x  – 5 A rational expression is an expression of the form where P and Q are polynomials and Q is not 0. A function defined by a rational expression is a rational function. The domain of a rational function is the set of values that can be used to replace the variable.

Tobey & Slater, Intermediate Algebra, 5e - Slide #4 Simplifying by Factoring Example: Find the domain of Set the denominator equal to 0. x 2 – 2x – 8 = 0 Factor. (x + 2)(x – 4) = 0 Use the zero factor property. x + 2 = 0 or x – 4 = 0 Solve for x. x = – 2x = 4 The domain of y = f(x) is all real numbers except – 2 and 4.

Tobey & Slater, Intermediate Algebra, 5e - Slide #5 Basic Rules of Fractions For any polynomials a, b, or c, where b and c  0. Example: Reduce.

Tobey & Slater, Intermediate Algebra, 5e - Slide #6 Simplifying by Factoring Example: Simplify. Factor 5 from the numerator. Apply the basic rule of fractions.

Tobey & Slater, Intermediate Algebra, 5e - Slide #7 Simplifying by Factoring Example: Simplify. Factor – 2 from the numerator. Apply the basic rule of fractions. Remember that when a negative number is factored from a polynomial, the sign of each term in the polynomial changes.

Tobey & Slater, Intermediate Algebra, 5e - Slide #8 Simplifying by Factoring Example: Simplify. Factor x from the numerator. Factor the numerator. Apply the basic rule of fractions. Factor the denominator.

Tobey & Slater, Intermediate Algebra, 5e - Slide #9 Multiplying Rational Expressions For any polynomials a, b, c, and d, where b and d  Rational expressions may be multiplied and then simplified. Rational expressions may also first be simplified and then multiplied. This method is usually easier.

Tobey & Slater, Intermediate Algebra, 5e - Slide #10 Simplifying the Product Example: Multiply. Factor each numerator and denominator. Apply the basic rule of fractions. Factor again whenever possible.

Tobey & Slater, Intermediate Algebra, 5e - Slide #11 Dividing Rational Expressions The definition for division of fractions is Example: Divide. 2 Invert the second fraction and multiply. This is called the reciprocal. Apply the basic rule of fractions.

Tobey & Slater, Intermediate Algebra, 5e - Slide #12 Simplifying the Quotient Example: Divide. Invert the second fraction and multiply. Apply the basic rule of fractions. Factor the numerator and denominator.