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Section 14.3 Least common denominators
1.
Slide - 1Copyright
© 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G 2 Rational Expressions and Applications 14
2.
Slide - 2Copyright
© 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G 1. Find the least common denominator for a list of fractions. 2. Write equivalent rational expressions. Objectives 14.3 Least Common Denominators
3.
Slide - 3Copyright
© 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Find the LCD for a List of Fractions Finding the Least Common Denominator (LCD) Step 1 Factor each denominator into prime factors. Step 2 List each different denominator factor the greatest number of times it appears in any of the denominators. Step 3 Multiply the denominator factors from Step 2 to find the LCD.
4.
Slide - 4Copyright
© 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Example Find the LCD for the fractions. 2 7 5 , 12 18x x Find the LCD for a List of Fractions Step 1 12x = 2 · 2 · 3 · x = 22 · 3 · x 18x2 = 2 · 3 · 3 · x2 = 2 · 32 · x2 Step 2 The greatest number of times that the factor 2 appears is twice. The greatest number of times both 3 and x appear is twice. Step 3 LCD = 22 · 32 · x2 = 36x2
5.
Slide - 5Copyright
© 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Example Find the LCD for the fractions. 2 2 4 5 , 15 3 6y y y Find the LCD for a List of Fractions 15y2 = 3 · 5 · y2 3y2 – 6y = 3 · y · (y – 2) LCD = 3 · 5 · y2 · (y – 2) = 15y2(y – 2)
6.
Slide - 6Copyright
© 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Example Find the LCD for fractions in the list. 2 2 2 2 2 5 1 , , 12 9 7 12 z z z z z z z Find the LCD for a List of Fractions z2 + z – 12 = (z + 4)(z – 3) z2 – 9 = (z + 3)(z – 3) z2 + 7z + 12 = (z + 4)(z + 3) LCD = (z + 4)(z – 3)(z + 3)
7.
Slide - 7Copyright
© 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Write Equivalent Rational Expressions Writing a Rational Expression with a Specified Denominator Step 1 Factor both denominators. Step 2 Decide what factor(s) the denominator must be multiplied by in order to equal the specified denominator. Step 3 Multiply the rational expression by that factor divided by itself. (That is, multiply by 1.)
8.
Slide - 8Copyright
© 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Example Write each rational expression as an equivalent expression with the indicated denominator. 3 7 35 ? x x Write Equivalent Rational Expressions First, factor the denominator on the right. Then compare the denominator on the left with the one on the right to decide what factors are missing. 3 5 7 5 x x x 2 15 35 x x Factors of 5 and x are missing. Multiply the numerator and denominator by 5x. 5 3 7 7 ? x x
9.
Slide - 9Copyright
© 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Example Write each rational expression as an equivalent expression with the indicated denominator. 3 2 17 (a) 3 4 24 32 ? y y y Write Equivalent Rational Expressions 2 2 17 3 4 8 8 y yy 2 3 2 136 24 32 y y y Factor the denominator on the right. The missing factor is 8y2. 2 17 3 4 8 (3 ? 4) y y y
10.
Slide - 10Copyright
© 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Example (cont) Write each rational expression as an equivalent expression with the indicated denominator. 2 3 2 3 (b) 16 16 ? 16 z z z z Write Equivalent Rational Expressions 2 3 1 1 16z z z Factor the denominator in each rational expression. The factor z + 1 is missing. ?3 4 4 1 4 4 z z z z z 3 2 3 3 16 16 z z z z
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