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This expression should be extremely easy to simplify, but for some reason I can't do it.

$$\frac{x^4-1}{x-1}$$

I know it simplifies down to this, but I don't know how to get there

$$x^3+x^2+x+1$$

This is a very basic question on my calculus worksheet, I would appreciate if anyone could explain how the first expression simplifies down to the second.

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Do you know the polynomial long division algorithm? Or the synthetic division abbreviation of it? – MJD May 30 '12 at 20:12
up vote 4 down vote accepted

Write it as the difference of two squares $$\frac{(x^2-1)(x^2+1)}{(x-1)}$$ $$\frac{(x+1)(x-1)(x^2+1)}{(x-1)}$$ $$(x+1)(x^2+1)=x^3+x^2+x+1$$

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To clarify Mark Dominus's comment, suppose that you don't know the expression on the right, but you want to simplify $\displaystyle{\frac{x^4-1}{x-1}}$ as much as possible by simplifying the numerator as much as possible. Well, you can't dive $x^4$ directly by $x-1$, but you know that $x^3\cdot(x-1)=x^4-x^3$, so add and subtract $x^3$: write the numerator as $x^4-1=(x^4-x^3)+(x^3-1)$ and your fraction as $\displaystyle{\frac{x^4-1}{x-1} = \frac{(x^4-x^3)+(x^3-1)}{x-1}=\frac{x^4-x^3}{x-1}+\frac{x^3-1}{x-1} = x^3+\frac{x^3-1}{x-1}}$. Now can you see how to keep going from here?

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very great explanation – dato datuashvili May 30 '12 at 20:50

you have $(x^2-1)(x^2+1)/(x-1)$,so $(x^2-1)=(x-1)(x+1)$,if we divide it by $(x-1)$,we will have $(x+1)(x^2+1)=x^3+x+x^2+1$

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You can use the polynomial long division method.

Stage 1

When you divide $x^4$ by $x$, you get $x^3$. When you multiply $x^3$ by $x - 1$, you get $x^4 - x^3$. After subtracting that from $x^4 - 1$, you get $x^3 - 1$. From there, you have $$x^3 + \frac {x^3 - 1}{x - 1}$$

Stage 2

You can apply the same method and get: $$x^2 + \frac {x^2 - 1}{x - 1}$$ After this stage, you get: $$x^3 + x^2 + \frac{x^2 - 1}{x - 1}$$ You will begin to notice a pattern. The degree gets smaller when you have to divide in every step.

Stage 3

Following the pattern, when you divide $x^2 - 1$ by $x - 1$, you get $$x + \frac {x - 1}{x - 1}$$ At the end of this stage, you get $$x^3 + x^2 + x + \frac {x - 1}{x - 1}$$

Stage 4

Following the pattern (in which this stage will be easy), you can use the $$\frac aa = 1$$ formula and get 1.

End of Problem

At the end, you will have $x^3 + x^2 + x + 1$. Hmm, something's weird here. In every stage you wen through, the degree got smaller. Also in the answer, the degree gets smaller! Anyways, the answer is $x^3 + x^2 + x + 1$.

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