# Simplifying Complex Fractions

• Jun 16th 2009, 05:12 AM
BG5965
Simplifying Complex Fractions
Hi, I have a fraction which I can't simplify fully.

http://i301.photobucket.com/albums/n...5/Noether3.jpg

I know that http://i301.photobucket.com/albums/nn74/BL5965/x2-4.jpg becomes (x + 2)(x - 2)

By solving with a calculator, the answer is:

http://i301.photobucket.com/albums/n...oether3Ans.jpg

How do I get there, with working?

Thanks, BG
• Jun 16th 2009, 06:05 AM
Quote:

Originally Posted by BG5965
Hi, I have a fraction which I can't simplify fully.

http://i301.photobucket.com/albums/n...5/Noether3.jpg

I know that http://i301.photobucket.com/albums/nn74/BL5965/x2-4.jpg becomes (x + 2)(x - 2)

By solving with a calculator, the answer is:

http://i301.photobucket.com/albums/n...oether3Ans.jpg

How do I get there, with working?

Thanks, BG

the numerator can be simplified to $\displaystyle x^2+1$

$\displaystyle 2x^2+x+4+\frac{1}{x}+\frac{2}{x^2}=0$

$\displaystyle 2x^2+\frac{2}{x^2}+x+\frac{1}{x}+4=0$

$\displaystyle 2(x^2+\frac{1}{x^2})+x+\frac{1}{x}+4=0$

Let $\displaystyle y=x+\frac{1}{x}$

$\displaystyle 2(y-2)+y+4=0$

$\displaystyle 3y=0$, $\displaystyle y=0$thus $\displaystyle x+\frac{1}{x}=0$which implies $\displaystyle x^2+1=0$

Anyways , i am not sure about the denominator . Sorry about that ..
• Jun 16th 2009, 06:57 AM
stapel
You can make the simplifying assumption that they've given you something "find-able" in the way of factors. (I'm sure you've checked already, and determined that the numerator has no real roots, and that the denominator has only irrational roots.)

Since the denominator has no linear term, then the constant terms and the linear terms in the factors must create terms that cancel out. So let's guess that the leading coefficients are "2", and that the constant terms are -2 and +2. Then let's see if we can find a product that works:

(2x^2 + ax + 2)(2x^2 + bx - 2)

= 4x^4 + (2a + 2b)x^3 + (ab)x^2 + (2b - 2a)x - 4

= 4x^4 + 4x^3 + x^2 - 4

Then 2b - 2a = 0, so 2b = 2a and thus a = b. This gives us:

4a = 4

...so a = 1. Checking:

a^2 = 1

...which matches the given coefficient of the squared term. Then the factors turn out to be:

(2x^2 + x + 2)
(2x^2 + x - 2)

Divide each of these into the numerator, and see which one comes out "even".

No, I don't see a way to do this in a more straightforward manner... (Wondering)