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

 Jan 11, 2018

Best Answer 

 #1
avatar+2341 
+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!

 Jan 12, 2018
edited by TheXSquaredFactor  Jan 12, 2018
 #1
avatar+2341 
+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
edited by TheXSquaredFactor  Jan 12, 2018
 #2
avatar+101870 
+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  + 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

 

 

cool cool cool

 

 

 

 

 

 

\(\)

.
 Jan 12, 2018
edited by CPhill  Jan 12, 2018

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