Partial Fractions

**HNCMATHS** · January 16th 2010, 12:38 PM

Hello,

Could someone please help me by telling me what the best way to resolve the following equation into three partial fractions is?

$6x^2 +5x -2 \div x(x-1)(2x+1)$

**e^(i*pi)** · January 16th 2010, 12:44 PM

Originally Posted by HNCMATHS

Hello,

Could someone please help me by telling me what the best way to resolve the following equation into three partial fractions is?

$6x^2 +5x -2 \div x(x-1)(2x+1)$

$\frac{6x^2+5x-2)}{x(x-1)(2x+1)} = \frac{A}{x} + \frac{B}{x-1} + \frac{C}{2x-1}$

Which becomes $6x^2+5x-2 = A(x-1)(2x+1) + B(x)(2x+1) + C(x)(x-1)$

It's best to find $f(0)$ , $f(1)$ and $f\left(-\frac{1}{2}\right) $

**HNCMATHS** · January 16th 2010, 12:48 PM

Sorry I do not understand your reply

**Archie Meade** · January 16th 2010, 01:46 PM

$\frac{A}{x}+\frac{B}{x-1}+\frac{C}{2x+1}=\frac{A(x-1)(2x+1)+Bx(2x+1)+Cx(x-1)}{x(x-1)(2x+1)}$

$=\frac{6x^2+5x-2}{x(x-1)(2x+1)}$

Now, you can ignore the denominators and multiply out the numerator on the right and compare terms with the left to find A, B and C.

The reason $e^{i*pi}$ recommended now finding f(0), f(1) and f(-0.5) is because f(0) finds A since the parts with B and C will be zero.

You get $6(0)+5(0)-2=A(-1)(1)$ and so on for B and C.

**Dinkydoe** · January 16th 2010, 01:47 PM

For more examples of partial fractions:

Partial fraction - Wikipedia, the free encyclopedia

**Soroban** · January 16th 2010, 03:17 PM

Helolo, HNCMATHS!

I'll show you the method I was taught.
I think it's much simpler; don't know why everyone doesn't teach it.

Resolve into partial fractions: . $\frac{6x^2 +5x -2}{x(x-1)(2x+1)}$

Set up the equation: . $\frac{6x+5x-2}{x(x-1)(2x+1)} \;=\;\frac{A}{x} + \frac{B}{x-1} + \frac{C}{2x+1}$

Multiply by the LCD: . $6x^2 + 5x - 2 \;=\;(x-1)(2x+1)A + x(2x+1)B + x(x-1)C$

Select some "good" values for $x\!:$

Let $x=0:$ . . . . . $0 + 0 - 2 \:=\:(\text{-}1)(1)A + 0\!\cdot\! B + 0\!\cdot\! C$ . . . . . $\Rightarrow$ . . . $\text{-}A \:=\:\text{-}2\quad\Rightarrow\quad\;\,\boxed{ A \:=\:2}$

Let $x = 1:$ . . . . $6+5-2 \:=\:0\cdot A + (1)(3)B + 0\cdot C$ . . . . . . $\Rightarrow\qquad 3B \:=\: 9$ . . $\Rightarrow\quad\;\boxed{ B\: =\: 3}$

Let $x = \text{-}\tfrac{1}{2}:\;\;6\left(\tfrac{1}{4}\right) + 5\left(\text{-}\tfrac{1}{2}\right)-2 \:=\: 0\cdot A + 0\cdot B + \left(\text{-}\tfrac{1}{2}\right)\left(\text{-}\tfrac{3}{2}\right)C \quad\Rightarrow$ . . . $\tfrac{3}{4}C \:=\:\text{-}3 \quad\;\;\Rightarrow\quad\boxed{ C \:=\:\text{-}4}$

Therefore: . $\frac{6x^2+5x-2}{x(x-1)(2x+1)} \;=\;\frac{2}{x} + \frac{3}{x-1} - \frac{4}{2x+1}$

[Edit: oops! .I see that $e^{i\pi}$ recommended this method.]

**Plato** · January 16th 2010, 03:45 PM

Originally Posted by HNCMATHS

Could someone please help me by telling me what the best way to resolve the following equation into three partial fractions is? $6x^2 +5x -2 \div x(x-1)(2x+1)$

Look at this result. It was done with a CAS costing about 40$ US.
There are even free programs, which will do this!
The idea of doing partial fractions has gone the way of square roots and horseshoes.
When has anyone done the square of 13 by hand?

**HNCMATHS** · January 17th 2010, 02:03 AM

Originally Posted by Archie Meade

$\frac{A}{x}+\frac{B}{x-1}+\frac{C}{2x+1}=\frac{A(x-1)(2x+1)+Bx(2x+1)+Cx(x-1)}{x(x-1)(2x+1)}$

$=\frac{6x^2+5x-2}{x(x-1)(2x+1)}$

Now, you can ignore the denominators and multiply out the numerator on the right and compare terms with the left to find A, B and C.

The reason $e^{i*pi}$ recommended now finding f(0), f(1) and f(-0.5) is because f(0) finds A since the parts with B and C will be zero.

You get $6(0)+5(0)-2=A(-1)(1)$ and so on for B and C.

***
Hello Archie, I understand what you have done up until the line starting... Now , you can ignore the denominators ??

Sorry

Chris

**HNCMATHS** · January 17th 2010, 02:09 AM

My main problem is that i dont understand how to work out that values that we need to plug in for $x$

EDIT**

I think I have worked it out, $x$ has to equal 0 in the divisor? so in our case where we have $2x+1$ ..... $x$ would equal $-0.5$

Its just clicked, again thank you all and i will be clicking thanks on all your posts.

**e^(i*pi)** · January 17th 2010, 02:20 AM

Originally Posted by HNCMATHS

My main problem is that i dont understand how to work out that values that we need to plug in for $x$

EDIT**

I think I have worked it out, $x$ has to equal 0 in the divisor? so in our case where we have $2x+1$ ..... $x$ would equal $-0.5$

Its just clicked, again thank you all and i will be clicking thanks on all your posts.

Glad to hear it

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