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9x+5= Express (5x-13)/((x-1)(x-3)^(2)) in the form (A)/((x-)

9x+5= 9 x+5= \newlineExpress 5x−13(x−1)(x−3)2 \frac{5 x-13}{(x-1)(x-3)^{2}} in the form A(x− \frac{A}{(x-}

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Full solution

Q. 9x+5= 9 x+5= \newlineExpress 5x−13(x−1)(x−3)2 \frac{5 x-13}{(x-1)(x-3)^{2}} in the form A(x− \frac{A}{(x-}
  1. Set up partial fraction decomposition: First, let's set up the partial fraction decomposition. We have (5x−13)/((x−1)(x−3)2)=A/(x−1)+B/(x−3)+C/(x−3)2(5x-13)/((x-1)(x-3)^{2}) = A/(x-1) + B/(x-3) + C/(x-3)^{2}.
  2. Clear fractions by multiplying: Now, we'll multiply both sides by the denominator to clear the fractions: (5x−13)=A(x−3)2+B(x−1)(x−3)+C(x−1)(5x-13) = A(x-3)^{2} + B(x-1)(x-3) + C(x-1).
  3. Expand and find AA, BB, CC: Next, we'll expand the right side to find AA, BB, and CC: (5x−13)=A(x2−6x+9)+B(x2−4x+3)+C(x−1)(5x-13) = A(x^2-6x+9) + B(x^2-4x+3) + C(x-1).
  4. Solve for CC by setting x=1x=1: Let's set x=1x=1 to solve for CC: (5(1)−13)=A(1−6+9)+B(1−4+3)+C(1−1);−8=4A+0B+0C;C=−2(5(1)-13) = A(1-6+9) + B(1-4+3) + C(1-1); -8 = 4A + 0B + 0C; C = -2.

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