Write the partial fraction decomposition of the rational expression.?

The denominator factors as x(x - 1)². This gives two linear factors, one of which is repeated. Remember that for a repeated linear factor, there has to be one term for each occurrence in the decomposition.

(x + 2)/x(x - 1)² = A/x + B/(x - 1) + C/(x - 1)². Multiply both sides through by the whole denominator to get x + 2 = A(x - 1)² + Bx(x - 1) + Cx Expanding the right and collecting like powers of x x + 2 = (A + B)x² + (-2A - B + C)x + A Equating like powers across both sides of the equation, we see that A + B = 0, -2A - B + C = 1, and A = 2. The first and third equations give A = 2, B = -2.

Subbing into the middle one gives C = 1 + B + 2A = 1 - 2 + 4 = 3. So the decomposition is 2/x - 2/(x - 1) + 3/(x - 1)². You can get a common denominator and collect that to double check for accuracy.

I will list out a general method I follow 1) Factorize the polynomials in the numerator and denominator. For the given expression, it is (x+2)/x(x-1)^2 2) Typically (for integration purposes), you will want to reduce this further. The expected answer is going to be of the form A/x + B/(x-1) + C/(x-1)^2 , where A, B and C are real numbers.

Adding the three fractions together, we get A(x-1)^2 + Bx(x-1) + Cx / x(x-1^2) solve for A, B and C you can see that A = 2 (A is the only variable that is not multiplied by x) since A =2. B=-2 because there is no x^2 term. The only way x^2 term can vanish is when Ax^2= Bx^2 that means C= 3 (since the x term should total to 1).

Solving for A, B and C can be done faster with practice.

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