Section 5.5 : Partial Fractions
6. Determine the partial fraction decomposition of each of the following expression.
7x2−17x+38(x+6)(x−1)2Show All Steps Hide All Steps
Start SolutionThe first step is to determine the form of the partial fraction decomposition. For this problem the partial fraction decomposition is,
7x2−17x+38(x+6)(x−1)2=Ax+6+Bx−1+C(x−1)2 Show Step 2The LCD for this expression is (x+6)(x−1)2. Adding the terms back up gives,
7x2−17x+38(x+6)(x−1)2=A(x−1)2+B(x+6)(x−1)+C(x+6)(x+6)(x−1)2 Show Step 3Setting the numerators equal gives,
7x2−17x+38=A(x−1)2+B(x+6)(x−1)+C(x+6) Show Step 4For this problem we can pick “good” values of x to determine only two of the three constants. Here is that work.
x=−6:x=1:392=A(−7)2=49A28=7C→A=8C=4 Show Step 5To get the remaining constant we can use any value of x and plug that along with the values of A and C we found in the previous step into the equation from Step 3.
It really doesn’t matter what value of x we pick as long as it isn’t x=−6 or x=1 since we used those in the previous step. The idea here is to pick a value of x that won’t create “large” or “messy” numbers, if possible. Good choices are often x=0 or x=1(which we can’t use for this problem as noted above), provided they weren’t used in the previous step of course.
For this problem x=0 seems to be a good choice. Here is the work for this step.
38=(8)(−1)2+B(6)(−1)+(4)(6)38=32−6B6=−6B⇒B=−1 Show Step 6The partial fraction decomposition is then,
\require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{7{x^2} - 17x + 38}}{{\left( {x + 6} \right){{\left( {x - 1} \right)}^2}}} = \frac{8}{{x + 6}} - \frac{1}{{x - 1}} + \frac{4}{{{{\left( {x - 1} \right)}^2}}}}}