Examples:

1.
Write (5x+1)/(x^2 - 1) as the sum of two rational expressions.

    We factor the x2 - 1 into (x-1)(x+1) so to get (5x+1)/(x^2 -1) = A/(x-1) + B/(x+1)

    Now start with the right hand side and combine those terms. We get

    A/(x-1) + B/(x+1) = [(A+B)x + (A-B)]/(x^2 -1)

    This implies that

    A/(x-1) + B/(x+) = [(A+B)x + (A-B)]/(x^2 - 1)

    Hence 5x + 1 = (A+B)x + (A-B). This means that A+B = 5 and A-B = 1. By solving these equations, we get A = 3 and B = 2. Therefore we get

    (5x + 1)/(x^2 - 1) = 3/(x-1) + 2/(x+1)

2.
Break (3 - 5x)/(x^2 - 4x + 4) into partial fractions.

    Now x2 - 4x + 4 = (x - 2)2 therefore we use

    (3-5x)/(x^2 - 4x + 4) = A/(x-2) + B/(x-2)^2

    Again we work with the right hand side to get

    A/(x-2) + B/(x-2)^2 = [Ax + (B-2A)]/(x^2 - 4x + 4)

    Therefore we get that Ax + (B-2A) = -5x + 3. Then A = -5 and B = -7. So we see that

    (3-5x)/(x^2 -4x+4) = -5/(x-2) -7/(x-2)^2

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