# Further mathematics:partial fractions?

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

2) (x-1)(x^2+x+1)/(x^2+1)(3x^2+4x+5) and

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

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1) (3x+2) / [(x+1)(x^2 +x +2)]

First note that x^2 +x +2 cannot be factored. (If it could be factored, then that would be the first step).

(3x+2) / [(x+1)(x^2 +x +2)] = A/(x+1) + (Bx+C) / (x^2 +x +2)

Bx+C is used on the second term because x^2 +x +2 is of the second degree, so the term on top must be one degree lower.

Now multiply both sides by (x+1)(x^2 +x +2)

3x + 2 = A(x^2 +x +2) + (Bx+C)(x+1)

This is true for any value of x.

If you let x = -1, you can find A.

3(-1) + 2 = A((-1)^2 +(-1) +2) + (B(-1) +C)(1-1)

-3 + 2 = A(1-1+2) + 0

-1 = A(2)

A = -1/2

Replace A with -1/2

3x + 2 = (-1/2)(x^2 +x +2) + (Bx+C)(x+1)

Now let’s try x = 0

3(0) + 2 = (-1/2)(0^2 +0 +2) + (B(0)+C)(0+1)

2 = (-1/2)(2) + C(1)

2 = -1 + C

C = 3

Replace C with 3.

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

Now pick a third value for x (one that you haven’t already used). Plug it in and solve for C.

After you’ve found C, go back to the equation

A/(x+1) + (Bx+C) / (x^2 +x +2)

and put in the values you just found for A, B, and C.