Chapter 6 Rational Expressions and Equations

6.1 Rational Expressions A rational expression is an algebraic fraction with a numerator and denominator that are polynomials. Ex. x2 - 4 is a rational expression with a denominator of 1. Non-permissible Values: Whenever solving a radical expression, you need to identify any values that must be excluded or that are non-permissible values. Non-permissible values are all values that make the denominator zero. Ex. in x + 2 you must exclude the value for which x - 3 = 0, which is x = 3 x - 3 Rational Expressions can be simplified by: factoring the numerator and the denominator determining non-permissible values dividing both the numerator and the denominator by all common factors

Example 1: Identify all the non-permissible values for 3x x(2x - 3) Determine the values for which x(2x - 3) = 0 x = 0 or 2x - 3 = 0 x = 3 The non-permissible values are 0 and 3 2 2 Example 2: Simplify the rational expression and state the non-permissible values. 3x - 6 2x2 + x - 10 = 3(x - 2) (x - 2)(2x + 5) = 3(x - 2) = 3 2x + 5 x ≠ - 5 , 2 2

6.2 Multiplying and Dividing Rational Expressions Multiplying rational expressions is similar to multiplying rational expressions. Factor each numerator and denominator. Identify any non-permissible values. Divide both numerator and denominator by any common factors to create a simplified expression. Dividing rational expressions is similar to dividing fractions Convert division to multiplication by multiplying by the reciprocal of the divisor When dividing, no denominator can be equal to zero.

Example 1: Multiply and state the non-permissible values Example 1: Multiply and state the non-permissible values. (x2 + 5x + 6)(x + 1) (x2 + x - 2)(x + 3) = (x + 2)(x + 3)・(x + 1) (x - 1)(x + 2)・(x+ 3) = x + 1 x ≠ 1, -2, -3 x - 1 Example 2: Divide and state the non permissible values. x2 + x - 6 ÷ x - 2 x2 + 2x - 15 x - 3 = (x - 2)(x + 3)・ (x - 3) (x + 5)(x - 3)・ (x - 2) = x + 3 x + 5 x ≠ -5, 2, 3

6.3 Adding and Subtracting Rational Expressions Rational expressions can be added and subtracted the same way that simple fractions are added and subtracted. You can either add or subtract with the same denominator by adding or subtracting their numerator. You can add or subtract rational expressions with unlike denominators after you have written each as an equivalent expression with a common denominator. More than one common denominator is always possible, it is often easier to use the lowest common denominator (LCD)

Example 1: Simplify 3x + 2 + x - 4 - 2x-1 In this case, the LCD is 24. 4 8 6 = 3x + 2 (6) + x-4 (3) - 2x - 1 (4) 4 (6) 8 (3) 6 (4) = 18x + 12 + 3x -12 - 8x - 4 24 24 24 = 21x - 8x - 4 13x + 4 24 24 24 Example 2: Simplify 7 - 1 The LCD is 30y2 10y 15y2 = 7 (3y) - 1 (2) 10y (3y) 15y2 (2) = 21y - 2 y ≠ 0 30y2

6.4 Rational Equations Equations that involve rational expressions are known as rational equations. They can be solved by performing the same operations on both sides. Working with a rational equation is similar to working with rational expressions. To Solve a Rational Equation: factor each denominator identify the non-permissible values multiply both sides of the equation by the lowest common denominator solve by isolating the variable on one side of the equation check your answers

Example: Solve the following equation and check your answer Example: Solve the following equation and check your answer. State the restrictions on the variable. 2 + 10 = 1 LCD is 3(x + 2)(x - 2) x2 - 4 6x + 12 x - 2 2 + 10 = 1 (x + 2)(x - 2) 6(x + 2) x - 2 2 (3(x + 2)(x - 2)) + 5 (3(x + 2)(x - 2)) = 1 (3(x + 2)(x - 2)) (x + 2)(x - 2) 3(x + 2) x - 2 6 + 5(x - 2) = 3(x + 2) 6 + 5x - 10 = 3x + 6 Check: 2x = 10 x = 5 LS: 2 + 10 RS: 1 21 42 5 - 2 = 4 + 10 = 1 ✓ 42 3 = 1 ✓ 3