# Partial fractions

1. Aug 27, 2005

### deryk

(5x^4-6x^3+31x^2-46x-20)/(2x^5-3x^4+10x^3-14x^2+5)

I got
it = 1/(2x+1) + 4.75/(x-1) + -2/(x-1)^2 + 8.75(x^2+5)

My working was several pages so Im not gonna post it. I was wondering if any of you know if that is right? Are there any geniuses on here who can do them in there head?

2. Aug 27, 2005

### VietDao29

Hmmm, I don't think it's correct.
You can check your answer by pluging x in and check if they are equal.
x = 0: The first one gives: -4.
While the second gives: 38.
So it's wrong...
Viet Dao,

3. Aug 27, 2005

### TD

Factoring the denominator should give:

$$2x^5 - 3x^4 + 10x^3 - 14x^2 + 5 = \left( {x - 1} \right)^2 \cdot \left( {x^2 + 5} \right) \cdot \left( {2x + 1} \right)$$

So propose the partial fractions:

$$\frac{A}{{\left( {x - 1} \right)^2 }} + \frac{B}{{x - 1}} + \frac{{Cx + D}}{{x^2 + 5}} + \frac{E}{{2x + 1}}$$

Now, rather than working it out and rearranging in powers of x to get a large 5x5 system, try choosing values of x that simplify the calculations, i.e. x's for which parts become zero (zero's of the denominators)

4. Aug 28, 2005

### lurflurf

Looks like you factored the denominator correctly
try the cover up shortcut, each highest order term can be found by substituting the a in to the function except for the factor that is zero. Lower order terms are found by subtracting off higher order terms. For terms of the form x^2+a^2 x^2 may be substituted for to avoid complex numbers, or they may be treated with the others using complex numbers.
factor denominator
(5x^4-6x^3+31x^2-46x-20)/[(2x+1)(x-1)^2(x^2+5)]
x=-1/2
(5a^4-6a^3+31a^2-46a-20)/[(2x+1)(a-1)^2(a^2+5)]|a=-1/2
=1/(2x+1) [you were right]
x=1 (order 2)
(5a^4-6a^3+31a^2-46a-20)/[(2a+1)(x-1)^2(a^2+5)]|a=1
=-2/(x-1)^2 [you were right]
x=1 (order 1)
(5a^3-a^2+30a-16)/[(2a+1)(x-1)(a^2+5)]|a=1
=3/(x-1) [you were wrong]
where the quotient without remainder is used
(5x^4-6x^3+31x^2-46x-20)=(x-1)(5x^3-x^2+30x-16)-36
x^2=-5
here to make things simple we substitute for a^2 instead of a. We multiply numerator and denominator by the conjugate of the denominator to rationalize. The general form that results is (ua+v)/(x^2-a^2) we replace a with x in the answer.
(5a^4-6a^3+31a^2-46a-20)/[(2a+1)(a-1)^2(x^2+5)]
(5a^4-6a^3+31a^2-46a-20)/[(2a^3-3a^2+1)(x^2+5)]|a^2=-5
(125+30a-155-46a-20)/[(-10a+15+1)(x^2+5)]
(-50-16a)(10a+16)/[(-10a+16)(10a+16)((x^2+5)]
(-756a/756)/(x^2+5)|a=x
-x/(x^2+5) [you were wrong]