Expert: Ahmed Salami - 4/15/2010

Question
Hello:

Having trouble factoring a repeated quadratic factor.

2x^2 + 7x + 4 / (x + 1)^3

As a partial fraction get the Lowest Common Denominator as: A(x + 1) + B(x + 1)^2 + C(x) = 2x^2 + 7x + 4

I factor a (-1) into A value and B value to negate them to (0)
therefore:
and I get (-1) = C(-1) therefore C = 1.

The correct answer is C = (-1) according to the book and wolframalpha.



Unfortunately the book and www.wolframalpha.com (a math site that does math algorithms both say I'm wrong.

It seems so simple and yet I get the wrong answer. Sigh. . .

Anyway: if it is not too much trouble I'd appreciate how to factor this partial fraction. I'm studying math over the internet right now because I'm living in China and don't have teachers avaliable. So any help would be appreciated.

The answer wolframalpha and the book gives is {2/(x + 1)} + {3/(x +1)^2} - {1/(x + 1)^3}

Also when I multiplied the Lowest Common Denominator back into the partial fraction it is also works out: the book is right and my calculation is wrong. But I don't understand how to factor out the denominator (x + 1)^3 and keep getting wrong answers.

Thanks
Will Sperry
Wenshan China

Answer
Hi Will,
I'm guessing you split your fraction incorrectly in the first place.
A denominator of the form (x + 1)³ requires you to split the partial fractions as
A/(x + 1) + B/(x + 1)² + C/(x + 1)³
So,
(2x² + 7x + 4)/(x + 1)³  =  A/(x + 1) + B/(x + 1)² + C/(x + 1)³
                        =  [A(x + 1)² + B(x + 1) + C]/(x + 1)³
and we end up with
2x² + 7x + 4 = A(x + 1)² + B(x + 1) + C     (Note that you had Cx instead of C)
at x = -1
2 - 7 + 4 = C
C = -1
Expanding the expression
2x² + 7x + 4 = A(x² + 2x + 1) + B(x + 1) - 1
2x² + 7x + 4 = Ax² + 2Ax + A + Bx + B - 1
2x² + 7x + 4 = Ax² + x(2A + B) + (A + B - 1)
Comparing both sides of the equation
A = 2
2A + B = 7
2(2) + B = 7
B = 3
Also
A + B - 1 = 2 + 3 - 1
         = 4
which further confirms the correctness.
Therefore,
(2x² + 7x + 4)/(x + 1)³  =  [2/(x + 1)] + [3/(x + 1)²] - [1/(x + 1)³]

Good luck with the study.

Regards