# Simplifying Equations with Polynomial denominators/numerators

1. Feb 5, 2012

### darshanpatel

1. The problem statement, all variables and given/known data

I uploaded a picture of the question because I didn't want to confuse people from the way it looks because it is pretty long.

http://tinypic.com/r/2itpm39/5

2. Relevant equations

-None-

3. The attempt at a solution

I went from the original equation and took an 'x' out of the first numerator and then got the reciprocal of the x^n and multiplied that through

1. Original Equation
2. (((x^n)-8)/((x^2n)+(2x^n)+1)) x ((x^2n)-(4x^n)-5)/(x^n)

I don't know what I can factor out from this point..

Last edited: Feb 5, 2012
2. Feb 5, 2012

### SammyS

Staff Emeritus
x2n = (xn)2

That in itself may help you. If not, then let u = xn and substitute that.

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3. Feb 5, 2012

### darshanpatel

Do you mean x^2n= (x^2)^n

Also as an example, if I took x^n-8 and did that, would it be:

((x)^3n)-(2)^3
and i dont know where to go from there

and for this one:

(x^2n)-(4x^n)-5
((x^2)^n)-((4x)^n)-5
what would i do next?

4. Feb 5, 2012

### eumyang

While that's true, SammyS was right to say
x2n = (xn)2

So if you were to make the substitution u = xn,
xn - 8 would become u - 8.
With the substitution, it would be
x2n - 4xn - 5
= (xn)2 - 4xn - 5
= u2 - 4u - 5

Now do the same kind of substitution in the first denominator. Doesn't that expression look familiar now? And can't you do something with u2 - 4u - 5?

5. Feb 5, 2012

### darshanpatel

I substituted all the x^n with y's and ended up with:

(((x^n)-5)((x^n)-2)((x^2n)+(2x^n)+4))/(x^n)((x^n)+1)

Is that all it can get factored and simplified?

6. Feb 5, 2012

### darshanpatel

Also, I think i will lose points by substituting, what is the way without it?

7. Feb 5, 2012

### SammyS

Staff Emeritus
((xn ) - 2) ((x2n ) + (2xn ) + 4) = x3n - 8 ≠ xn - 8

8. Feb 7, 2012

### darshanpatel

I redid it and got: (((x^n)-8)((x^n)-5))/(x^n)((x^n)+1)

Is that correct?

Sorry, I didn't get what you are trying to say

9. Feb 7, 2012

### eumyang

If you mean
$$\frac{(x^n-8)(x^n-5)}{x^n(x^n+1)}$$
... then yes, looks right.

10. Feb 7, 2012

### Mentallic

Substituting isn't necessary, but makes things look simpler.

For example, if you were asked to factorize

$$x^2+4ax+4a^2$$

then it isn't very clear what needs to be done, but if you let 2a=b then you have

$$x^2+2bx+b^2$$ and this is clearly a perfect square, so you can factorize it as so:

$$(x+b)^2$$ and then you can substitute back in at the end to obtain

$$(x+2a)^2$$