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695. The fraction is the sum of .

Set the original fraction equal to the sum. Now add the three fractions on the right together. Let the numerator on the left equal the numerator on the right.

x2 – 12x – 49

= Ax2 – 3Ax – 10A + Bx2 – 6Bx + 5B + Cx2 + Cx – 2C

Factor on the right.

–x2 + 12x – 3

= x2(A + B + C) + x(–3A – 6B + C) – 10A + 5B – 2C

Write a system of equations involving the coefficients of the x terms and the constants. Solve the system:  From the third equation, you find that C = –3. Substitute that into the second equation:

–9 = –3B + 4(–3)

–9 = –3B – 12

3 = –3B, so B = –1.

Then, solve for A in the first equation:

1 = A + (–1) + – 3, A = 5

The decomposition is .

696.     ...

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