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Rational Expressions and Equations: Simplifying, Multiplying, and Dividing

Learn to simplify, multiply, and divide rational expressions to solve equations and simplify algebraic expressions in this chapter. Understand domain restrictions and essential fraction rules.

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Rational Expressions and Equations: Simplifying, Multiplying, and Dividing

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  1. Rational Expressions and Equations Chapter 6

  2. § 6.1 Simplifying, Multiplying, and Dividing

  3. A rational expression is an expression of the form where P and Q are polynomials and Q is not 0. Fractional algebraic expression x – 5 Rational Expressions or 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.

  4. Simplifying by Factoring Example: Find the domain of x2 – 2x – 8 = 0 Set the denominator equal to 0. (x + 2)(x – 4) = 0 Factor. x + 2 = 0 or x – 4 = 0 Use the zero factor property. x = – 2 x = 4 Solve for x. The domain of y = f(x) is all real numbers except – 2 and 4.

  5. Example: Reduce. Basic Rules of Fractions Basic Rules of Fractions For any polynomials a, b, or c, where band c  0.

  6. Simplifying by Factoring Example: Simplify. Factor 5 from the numerator. Apply the basic rule of fractions.

  7. Remember that when a negative number is factored from a polynomial, the sign of each term in the polynomial changes. Simplifying by Factoring Example: Simplify. Factor –2 from the numerator. Apply the basic rule of fractions.

  8. Simplifying by Factoring Example: Simplify. Factor x from the numerator. Factor the numerator. Factor the denominator. Apply the basic rule of fractions.

  9. Multiplying Rational Expressions For any polynomials a, b, c, and d, where band d  0. Rational expressions may be multiplied and then simplified. Rational expressions may also first be simplified and then multiplied. 7 1 5 1 This method is usually easier.

  10. Simplifying the Product Example: Multiply. Factor each numerator and denominator. Factor again whenever possible. Apply the basic rule of fractions.

  11. Example: Divide. Dividing Rational Expressions The definition for division of fractions is Invert the second fraction and multiply. This is called the reciprocal. 2 Apply the basic rule of fractions.

  12. Simplifying the Quotient Example: Divide. Invert the second fraction and multiply. Factor the numerator and denominator. Apply the basic rule of fractions.

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