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FILE 17 Rational Expressions Math 0096 0097 Summer 2010

FILE 17 Rational Expressions Math 0096 0097 Summer 2010 -...

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[FILE 17] SOME ASPECTS ON RATIONAL EXPRESSIONS (Pages 621-653) Sometimes we will encounter mathematical expressions containing fractions and we must be able to simplify them. There are several techniques that can be used to simplify such an expression. Now let us examine some of the concepts under the above heading. I. REDUCING RATIONAL EXPRESSIONS Note: To reduce rational expressions is to divide; both, the numerator and the denominator by the same non-zero number an even number of times. Do not convert an improper fraction to a mixed number unless you are told to do so. Sometimes rational expressions involve polynomial expressions in which case you would factor, if possible, then reduce by cancellation. You MUST know the rules on FACTORING . Examples: Reduce 1) 24 = 2 Hint: To reduce, divide, both, the numerator and the denominator 36 3 by 12. 2) x(x+2) = x Hint: Since the given expressions are already in factored form, y(x+2) y reduce by canceling like quantities, if possible. 3) x 2 +5x+2 = (x+3)(x+2) = x+3 Hint: When factoring, always look for a common x 2 -4 (x+2)(x-2) x-2 monomial factor, first. If not, factor, and apply the cancellation process to reduce. 4) 4x 2 (6x 2 -13xy+6y 2 ) = 4xx(3x-2y)(2x-3y) = 2x(3x-2y) 6x(2x 2 +5xy-12y 2 ) 6x(2x-3y)(x+4y) 3(x+4y) TRY # 5 AND # 6 ON YOUR OWN: 5) 6x 2 -54 = 2x 2 +8x+6 6) 2x 3 +20x 2 +50x = 4x 3 -100x ANSWERS: 5) 3(x-3) 6) x+5 x+1 2(x-5) NOTE: Leave the result(s) in factored form as in examples 5 and 6 above. Exercise: Do problems 3,11,17,21,27,31,36,41,47,55,61, (Pages 630-632) II. Multiplication/Division with Rational Expressions See pages 633-638. 1
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NOTES: When multiplying or dividing rational expressions, i. factor where possible ii. if dividing, invert the divisor and multiply iii. use the cancellation process to reduce and simplify Examples: Perform the indicated operation. 1) 2 18 7 = 1 2) 9b 28c = 4 3) 4-x 2 7 = (2+x)(2-x) 7 = 2+x 9 35 4 5 7c 81b 9 14 2-x 14 (2-x) 2 4) x 2 +3x+2 x+3 = (x+2)(x+1) (x+3) =1 x+1 x 2 +5x+6 (x+1) (x+3)(x+2) 5) 3 ÷ 11 = 3 14 = 6 6) c 2 ÷ c = (c)(c) d = c 7 14 7 11 11 d d d c 7) x 2 -y ÷ (x-y) = (x+y)(x-y) 1 = x+y 8) 2x+4 ÷ 4x+8 = 2(x+2) 5(x+3) = 5 xy xy (x-y) xy 3x+9 5x+15 3(x+3) 4(x+2) 6 9) x 2 +4x+3 ÷ x+3 = (x+3)(x+1) (x-5) =1
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