Explanation Calculation: Consider the expression, 1 4 x 2 − 9 Note that the degree of an expression is the highest power of the variable in the expression. Here, numerator has degree 0 and denominator has degree 2. Since the degree of the expression written in the numerator is less than the degree of the expression written in the denominator so, the given rational expression is a proper fraction. Use the algebraic identity, u 2 − v 2 = u − v u + v to make the factors as shown below: 4 x 2 − 9 = 2 x 2 − 3 2 = 2 x − 3 2 x + 3 Consider that the degree of the factors 2 x − 3 and 2 x + 3 is 1 , which is linear. Since the two factors 2 x − 3 and 2 x + 3 are linear, write one partial fraction with numerator as an unknown constant for each of the linear factors. Thus, 1 4 x 2 − 9 = A 2 x − 3 + B 2 x + 3 The least common denominator (LCD) of the terms on the right-hand side is 2 x − 3 2 x + 3 = 4 x 2 − 9 . So, make the denominators same. A 2 x − 3 + B 2 x + 3 = A 2 x + 3 2 x − 3 2 x + 3 + B 2 x − 3 2 x − 3 2 x + 3 = A 2 x + 3 + B 2 x − 3 4 x 2 − 9 Thus, 1 4 x 2 − 9 = A x + 1 + B x − 1 4 x 2 − 9 . Now, the denominators on the either side are same. So, multiply both sides by the LCD to get the basic equation. 1 = A 2 x + 3 + B 2 x − 3 To solve the basic equation for the unknowns A and B , substitute real numbers to make the distinct linear factors equal to zero. To calculate the value of A , substitute 2 x − 3 = 0 , that is, x = 3 2 in the basic equation. For x = 3 2 , 1 = A 2 3 2 + 3 + B 2 3 2 − 3 1 = A 3 + 3 + B 3 − 3 = 6 A + 0 1 6 = A Thus, A = 1 6 . To calculate the value of B , substitute 2 x + 3 = 0 , that is, x = − 3 2 in the basic equation. For x = − 3 2 , 1 = A 2 − 3 2 + 3 + B 2 − 3 2 − 3 1 = A − 3 + 3 + B − 3 − 3 1 = 0 − 6 B − 1 6 = B Thus, B = − 1 6

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Chapter 6.4, Problem 22E

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To calculate: The form of partial fraction decomposition of the rational expression 14x2−9 and check the answer algebraically.

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Chapter 6.4, Problem 21E

Chapter 6.4, Problem 23E

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The form of partial fraction decomposition of the rational expression 1 4 x 2 − 9 and check the answer algebraically.

Solution Summary: The author calculates the form of partial tion decomposition of the rational expression 14x2-9 and checks the answer algebraically.

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