Simplify $\frac{x^2+2x^4+3x^6+...+1005x^{2010}}{2x+4x^3+6x^5+...+2010x^{2009}}$.

Guest Jan 11, 2018

#1**+1 **

\(\frac{x^2+2x^4+3x^6+...+1005x^{2010}}{2x+4x^3+6x^5+...+2010x^{2009}}\)

I see that both the numerator and denominator both have a GCF. The numerator has a GCF of *x^2*, and the denominator has a GCF of *2x*. Let's factor that out.

\(\frac{x^2(1+2x^2+3x^4+...+1005x^{2008})}{2x(1+2x^2+3x^4+...+1005x^{2008})}\)

As you can see here, the sequence in the middle cancels out! That leaves us with a simplified expression.

\(\frac{x^2}{2x}\)

Don't stop yet! This fraction can be simplified even more to \(\frac{x}{2}\) because of the common term of x. Wow! I never would have guessed that the monstrosity would simplify to something nice!

TheXSquaredFactor
Jan 12, 2018

#1**+1 **

Best Answer

\(\frac{x^2+2x^4+3x^6+...+1005x^{2010}}{2x+4x^3+6x^5+...+2010x^{2009}}\)

I see that both the numerator and denominator both have a GCF. The numerator has a GCF of *x^2*, and the denominator has a GCF of *2x*. Let's factor that out.

\(\frac{x^2(1+2x^2+3x^4+...+1005x^{2008})}{2x(1+2x^2+3x^4+...+1005x^{2008})}\)

As you can see here, the sequence in the middle cancels out! That leaves us with a simplified expression.

\(\frac{x^2}{2x}\)

Don't stop yet! This fraction can be simplified even more to \(\frac{x}{2}\) because of the common term of x. Wow! I never would have guessed that the monstrosity would simplify to something nice!

TheXSquaredFactor
Jan 12, 2018

#2**+1 **

\(\frac{x^2+2x^4+3x^6+...+1005x^{2010}}{2x+4x^3+6x^5+...+2010x^{2009}}\)

Note that we can factor the numerator as

x (x + 2x^{3} + 3x^{5} +....+ 1005x^{2009} ) (1)

And note that we can factor the denominator as

2 (x + 2x^{3} + 3x^{5} + ....+ 1005x^{2009} ) (2)

So

( 1 ) / (2) will result in x / 2

\(\)

CPhill
Jan 12, 2018