Final answer:
The student is asked to find the partial fraction decomposition of (x-12)/(x²-4x). The process involves factoring the denominator and setting up the equivalent sum of simpler fractions, then solving for the constants to complete the decomposition.
Step-by-step explanation:
The question is asking for the partial fraction decomposition of the rational function (x-12)/(x²-4x). This process involves breaking down a complicated fraction into simpler fractions that add up to the original. To start, we factor the denominator which is x(x-4). Then we set up the partial fraction as:
A/x + B/(x-4) = (x-12)/(x(x-4))
Multiplying both sides by the common denominator x(x-4), we get:
A(x-4) + Bx = x - 12
By matching coefficients on both sides of the equation, we can solve for A and B. Once we determine the values of A and B, they are substituted back into the partial fractions to get the final decomposed form.