Expressing in partial fractions

**erika** · July 7th 2007, 11:11 AM

$(9x^2) / [(x -1)^2 * (x + 2)]$

It's part of a set of partial fractions I have to do.. I can't seem to solve this?

**Jhevon** · July 7th 2007, 11:13 AM

Originally Posted by erika

$(9x^2) / [(x -1)^2 * (x + 2)]$

It's part of a set of partial fractions I have to do.. I can't seem to solve this?

$\frac {9x^2}{(x - 1)^2 (x + 2)} = \frac {A}{x - 1} + \frac {B}{(x - 1)^2} + \frac {C}{x + 2}$

Can you take it from here?

**erika** · July 7th 2007, 11:15 AM

Meh I did that and then tried substituting values for $x$ but it didn't seem to work

**Jhevon** · July 7th 2007, 11:18 AM

Originally Posted by erika

Meh I did that and then tried substituting values for $x$ but it didn't seem to work

Substitute values of x? do you know the method of partial fractions?

multiply through by the denominator of the left side, we get:

$9x^2 = A(x - 1)(x + 2) + B(x + 2) + C(x - 1)^2$

Can you take it from here? You can multiply out and equate coefficients, or you can choose convenient values of x that wipes out two variables at a time

**erika** · July 7th 2007, 11:35 AM

sub x: 1
9 = 3B
3 = B

sub x: -1
36 = 9C
4 = C

By comparing coefficient of x^2:
9 = A + C
A = 5

Is that correct?

**Jhevon** · July 7th 2007, 11:41 AM

Originally Posted by erika

sub x: 1
9 = 3B
3 = B

sub x: -1
36 = 9C
4 = C

By comparing coefficient of x^2:
9 = A + C
A = 5

Is that correct?

Yes! Good Job!

**erika** · July 7th 2007, 11:58 AM

Thank you!
One last question...

$ \frac {17x^2+23x+12}{(3x+4)(x^2+4)} = \frac{A}{3x+4} + \frac{B}{x^2+4} $

By cover-up rule, I got
$ A = \frac {17(-4/3)^2+23(-4/3)+12}{(-4/3)^2+4} = 2 $

But since $(x^2 +4)$ isn't linear, does it mean I follow the steps you mentioned earlier (multiply through by denominator, etc)?

**Jhevon** · July 7th 2007, 12:01 PM

Originally Posted by erika

Thank you!
One last question...

$ \frac {17x^2+23x+12}{(3x+4)(x^2+4)} = \frac{A}{3x+4} + \frac{B}{x^2+4} $

By cover-up rule, I got
$ A = \frac {17(-4/3)^2+23(-4/3)+12}{(-4/3)^2+4} = 2 $

what is this "cover-up" rule you are referring to?

But since $(x^2 +4)$ isn't linear, does it mean I follow the steps you mentioned earlier (multiply through by denominator, etc)?

yes, but you have to change the numerator. The numerator must always be one degree less than the denominator. so for the part where the denominator is $x^2 + 4$ you must write:

$\frac {Bx + C}{x^2 + 4}$

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