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# Algebraic Manipulation Question

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

Aug 9, 2022

#2
+2446
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Factor out x from the numerator of the fraction: $$x(x + 2x^3 + 3x^5 + \cdot \cdot \cdot + 1005x^{2009})$$

Now, factor out 2 from the denominator: $$2(x + 2x^3 + 3x^5 + \cdot \cdot \cdot + 1005x^{2009})$$

So we have $${x(x + 2x^3 + 3x^5 + \cdot \cdot \cdot + 1005x^{2009}) \over 2(x + 2x^3 + 3x^5 + \cdot \cdot \cdot + 1005x^{2009})} = \color{brown}\boxed{x \over 2}$$

Aug 9, 2022

#1
+251
-1

Suppose that the expression equals y.

Dividing by x gives us, (x^2 + 2x^4 +... + 1005x^(2010))/(2x^2 + 4x^4 +... + 2010x^(2010)). This gives us a value of 1/2.

1/2 divided by x gives us $$\frac{1}{2x}$$

Aug 9, 2022
#2
+2446
+1

Factor out x from the numerator of the fraction: $$x(x + 2x^3 + 3x^5 + \cdot \cdot \cdot + 1005x^{2009})$$

Now, factor out 2 from the denominator: $$2(x + 2x^3 + 3x^5 + \cdot \cdot \cdot + 1005x^{2009})$$

So we have $${x(x + 2x^3 + 3x^5 + \cdot \cdot \cdot + 1005x^{2009}) \over 2(x + 2x^3 + 3x^5 + \cdot \cdot \cdot + 1005x^{2009})} = \color{brown}\boxed{x \over 2}$$

BuilderBoi Aug 9, 2022