Purchase MAT 104 - Chapter R Section 7 - Rational Expressions

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Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20R.7 Rational ExpressionsChapter R Section 7: Rational ExpressionsIn this section, we will… Reduce a Rational Expression to Lowest Terms Multiply and Divide Rational Expressions Add and Subtract Rational Expressions Simplify Complex Rational Expressionspunk mathematiciansR.7 Rational ExpressionsRational Expression – the quotient of two polynomialsexamples:What is the domain of each of the rational expressions above?335xx+-323 2 13 2x xx x+ ++ +FRACTIONSR.7 Rational Expressions: Reduce a Rational Expression to Lowest TermsSimplifying Rational Expressions: •Ensure that the numerator and denominator of the rational expression have no common factors (other than 1)•We will accomplish this by fully factoring both the numerator and denominator then cancelling out common factorsFraction Review:To reduce a fraction, you cancel common factors but we cannot cancel terms!36=23=R.7 Rational Expressions: Reduce a Rational Expression to Lowest TermsExamples: Reduce each rational expression to lowest terms.24 812 24x xx++22 5 31 2x xx+ --222 26x xa x ax x+ + ++ -You may leave your answers in factored-form.R.7 Rational Expressions: Multiply and Divide Rational ExpressionsMultiplying Rational Expressions: •Fully factor each numerator and denominator•Cancel any common factors (as long as one factor is in a numerator and the other is in the denominator…you can cancel them!)•Write as a single rational expression Fraction Review:20 735 8�You may leave your answers in factored-form.R.7 Rational Expressions: Multiply and Divide Rational ExpressionsExamples: Perform the indicated operation and simplify the result. You may leave your answer in factored form.2 26 93x x xx x- +�-2 22 26 64 9x x x xx x- - + -�- -3212 14 2xx x x+�+ -R.7 Rational Expressions: Multiply and Divide Rational ExpressionsDividing Rational Expressions: •Multiply the first fraction by the reciprocal of the second•Fully factor each numerator and denominator•Cancel any common factors •Write as a single rational expressionFraction Review:7219�R.7 Rational Expressions: Multiply and Divide Rational ExpressionsExamples: Perform the indicated operation and simplify the result. You may leave your answer in factored form.( )3421b bbb-� --322812 42 2xxx xx++- +-R.7 Rational Expressions: Add and Subtract Rational ExpressionsAdding and Subtracting Rational Expressions with Equal Denominators•Combine the numerators and place the result over the common denominator•Fully factor each numerator and denominator•Cancel any common factors •Write as a single rational expressionExample: Perform the indicated operation and simplify the result. Fraction Review:To add and subtract fraction, we need a common denominator.3 116 16+3 23 3x xx x+-- -R.7 Rational Expressions: Add and Subtract Rational ExpressionsAdding and Subtracting Rational Expressions with Unequal Denominators Finding a Common Denominator•Factor each denominator•Take the product of each of these factors (raised to a power equal to the greatest number of times that the factor appears in the polynomials)Example: Find the least common multiple (LCM) for the given polynomials.Fraction Review: finding the least common denominator (LCD).8 59 12-2 212 , 8 16x x x x- - - +2 23 27 , 2 15x x x- - -R.7 Rational Expressions: Add and Subtract Rational ExpressionsAdding and Subtracting Rational Expressions with Unequal Denominators•Factor each denominator•Take the product of each of these factors (raised to a power equal to the greatest number of times that the factor appears in the polynomials)•Combine the numerators and place the result over the common denominator•Fully factor each numerator (the denominators are already factored)•Cancel any common factors •Write as a single rational expressionFraction Review:To add and subtract fraction, we need a common denominator.8 59 12-{Finding LCD{SimplifyingR.7 Rational Expressions: Add and Subtract Rational ExpressionsExamples: Perform the indicated operation and simplify the result. You may leave your answer in factored form.61 1xx x-- -2 3 2 11 1x xx x- +-- +R.7 Rational Expressions: Add and Subtract Rational ExpressionsExample: Perform the indicated operation and simplify the result. You may leave your answer in factored form.213 5 24x xx x x+-- + -LCD=R.7 Rational Expressions: Add and Subtract Rational ExpressionsExample: Perform the indicated operation and simplify the result. You may leave your answer in factored form.2 22 6( 2) ( 1) ( 2)( 1)x x x x-+ - + -LCD=R.7 Rational Expressions: Add and Subtract Rational ExpressionsExample: Perform the indicated operation and simplify the result. You may leave your answer in factored form.22 62 2a a a++ + -LCD=R.7 Rational ExpressionsExample: Reduce the expression below to lowest terms.2(4 1) 5 (5 2) 4(5 2)x xx+ �- - �-Recall that we can only cancel factors (not terms) – we must perform the indicated operations to the numerator as it is not in factored form.R.7 Rational Expressions: Simplify Complex Rational ExpressionsSimplifying Complex Rational Expressions: A complex fraction is a fraction that has a rational expression (i.e. fraction) in the numerator or the denominator or both.•Write the numerator and denominator of the complex fraction as a single rational expression (fraction)•Divide the fractions•Simplify the resultFraction Review:To simplify a complex fraction means to express it as a single fraction. 236abacbR.7 Rational Expressions: Simplify Complex Rational ExpressionsExample: Perform the indicated operation and simplify the result. You may leave your answer in factored form.1 11 1x yx y+-R.7 Rational Expressions: Simplify Complex Rational ExpressionsExample: Perform the indicated operation and simplify the result. You may leave your answer in factored form.1112xxxx-+--R.7 Rational ExpressionsIndependent Practice You learn math by doing math. The best way to learn math is to practice, practice, practice. The assigned homework examples provide you with an opportunity to practice. Be sure to complete every assigned problem (or more if you need additional practice). Check your answers to the odd-numbered problems in the back of the text to see whether you have correctly solved each problem; rework


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