Simplify $\frac{x^2+2x^4+3x^6+...+1005x^{2010}}{2x+4x^3+6x^5+...+2010x^{2009}}$.
\(\frac{x^2+2x^4+3x^6+...+1005x^{2010}}{2x+4x^3+6x^5+...+2010x^{2009}}\)
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!
\(\frac{x^2+2x^4+3x^6+...+1005x^{2010}}{2x+4x^3+6x^5+...+2010x^{2009}}\)
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!
\(\frac{x^2+2x^4+3x^6+...+1005x^{2010}}{2x+4x^3+6x^5+...+2010x^{2009}}\)
Note that we can factor the numerator as
x (x + 2x3 + 3x5 +....+ 1005x2009 ) (1)
And note that we can factor the denominator as
2 (x + 2x3 + 3x5 + ....+ 1005x2009 ) (2)
So
( 1 ) / (2) will result in x / 2
\(\)