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# Help asap

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

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!

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