Decompose P/Q, Where Q Has Repeated Linear Factors
Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the
constants.
-$$\frac { 9 x + 6 } { ( x - 6 ) ^ { 2 } }$$
A) $\frac { 9 } { x - 6 } + \frac { 60 } { ( x - 6 ) ^ { 2 } }$
B) $\frac { 9 } { x - 6 } + \frac { 108 } { ( x - 6 ) ^ { 2 } }$
C) $\frac { 9 } { x - 6 } + \frac { x + 60 } { ( x - 6 ) ^ { 2 } }$
D) $\frac { 1 } { x - 6 } + \frac { x + 60 } { ( x - 6 ) ^ { 2 } }$

Question

Quizplus · Accepted Answer

The denominator has a repeated linear factor, so the partial fraction decomposition will have the form:$$\frac { A } { x - 6 } + \frac { B } { ( x - 6 ) ^ { 2 } }$$Multiplying both sides by (x - 6)^2, we get:$$9x + 6 = A(x - 6) + B$$Setting x = 6, we get:$$9(6) + 6 = A(6 - 6) + B$$$$54 + 6 = 0 + B$$$$B = 60$$Substituting B = 60 into the equation, we get:$$9x + 6 = A(x - 6) + 60$$Setting x = 0, we get:$$9(0) + 6 = A(0 - 6) + 60$$$$6 = -6A + 60$$$$-6A = 54$$$$A = -9$$Therefore, the partial fraction decomposition of the rational expression is:$$\frac { -9 } { x - 6 } + \frac { 60 } { ( x - 6 ) ^ { 2 } }$$

Decompose P/Q, Where Q Has Repeated Linear Factors Write the Form

Decompose P/Q, Where Q Has Repeated Linear Factors
Write the Form