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M098 Carson Elementary and Intermediate Algebra 3e Section 7.1 Objectives 1. 2. 3. 4. Evaluate rational expressions Find numbers that cause a rational expression to be undefined. Simplify rational expressions containing only monomials. Simplify rational expressions containing multiterm polynomials. Vocabulary Rational expression An expression that can be written in the form P , where P and Q are polynomials and Q Q ≠ 0. Prior Knowledge Rational numbers are numbers that can be expressed in the form a/b where a and b are integers and b≠0. Fractions are undefined if the denominator equals 0. To evaluate means to replace the variable(s) with the given value(s). New Concepts 1. Evaluate rational expressions. This chapter deals with rational expressions and rational equations. The word association for rational is fraction. We’ll spend some time reviewing our basic fraction arithmetic skills. We use exactly the same process with rational expressions (algebraic fractions) as we do with arithmetic fractions - the problems are just a little uglier looking!!! To evaluate a rational expression, replace the variable(s) with the given values. Example 1: Evaluate 3(2) 2 4(2) 5 3x 2 when x = 2. 4x 5 Replace the variable with the given number. 62 85 8 3 Example 2: Evaluate 4(3) (3 ) 3 12 0 V. Zabrocki 2011 Evaluate the numerator and the denominator. 4x when x = 3. x3 Replace the variable with the given number. Evaluate the numerator and the denominator. This expression is undefined. page 1 M098 Carson Elementary and Intermediate Algebra 3e Section 7.1 2. Find numbers that cause a rational expression to be undefined. One of the first concepts in Chapter 1 was that a fraction can not have 0 in the denominator. It is undefined when that occurs. The numerator (top) can be 0 but not the denominator. The same is true in this chapter. Since rational expressions have variables in the denominator, we have to continually remind ourselves to check to see which numbers will make the fraction undefined - in other words, when will the denominator equal 0. Example 3: When will x2 be undefined? x5 This will be undefined when the denominator (x – 5) equals 0. x–5=0 Solve the linear equation. x=5 The fraction will have 0 in the denominator if x = 5. The rational expression is undefined when x = 5. Example 4: When will x2 4 x 2 7x 10 be undefined? 2 This will be undefined when the denominator (x – 7x + 10) equals 0. 2 x - 7x + 10 = 0 Solve the quadratic equation. There should be 2 solutions. (x - 2) (x - 5) = 0 x-2=0 x-5=0 x=2 x=5 The fraction will have 0 in the denominator if x = 2 or x = 5. The rational expression is undefined when x = 2 or x = 5. Example 5: When will 2p 3 p 2 6p be undefined? 2 This will be undefined when the denominator (p + 6p) equals 0. 2 p + 6p = 0 Solve the quadratic equation. There should be 2 solutions. p(p + 6) = 0 p=0 p+6=0 p = -6 The fraction will have 0 in the denominator if p = 0 or p = -6. The rational expression is undefined when p = 0 or p = -6. V. Zabrocki 2011 page 2 M098 Carson Elementary and Intermediate Algebra 3e Section 7.1 3. Simplify rational expressions. In working with fractions in arithmetic, we always reduced them to lowest terms by dividing the numerator and denominator by the greatest common factor. If you remember back to when you first learned to reduce fractions, the process went something like this: 18 233 3 24 234 4 We can reduce these fractions because 2 and 3 are common factors. This is another one of those red flag areas. We can only reduce when we are multiplying - never when we add. 23 5 3 which does not equal . 25 7 5 Terms can NOT be reduced – only factors. Example 6: Simplify 14h 3k 21 h 27hhhk 37h Rewrite in factored form. 27hhhk 37h Reduce common factors. 2h 2k 3 Multiply remaining factors. Example 7: Simplify x 2 10x 25 x 2 25 . x 5x 5 x 5x 5 Rewrite in factored form. x 5x 5 x 5x 5 Reduce common factors. x5 x5 Example 8: Simplify by ay bx ax x 2 ax xy ay . b ay x x ax y Rewrite in factored form. b ay x x ax y Reduce common factors. ba xa V. Zabrocki 2011 page 3 M098 Carson Elementary and Intermediate Algebra 3e Example 9: Simplify Section 7.1 x 3 64 . 4x x 4x 2 4x 16 Rewrite in factored form. 4x x 4x 2 4x 16 x 4 x 4x 2 4x 16 x 4 x 2 4x 16 1 1 x 2 4x 16 (x – 4) and 4 – x are inverses. Factor out -1 so from one of them so the binomials match. Reduce common factors. It is common practice to remove the minus from the denominator. The -1 can be moved to the numerator. Distribute the -1. x 2 4x 16 V. Zabrocki 2011 page 4