Factorization of polynomials

Could someone explain to lớn me how the summation of the the geometric series explains the factorization?


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The long parenthesized term is a geometric series with first term $a^n-1$ and ratio $frac ba$ so mix $x=frac ba$


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I see the answer is accepted. But for future reference, another proof would be

Let $p(x)=x^n-a^n$. Clearly, $x=a$ is a solution. This means $x-a$ is a factor of $x^n-a^n.$

It is just a matter of simple polynomial division aafter that and so dividing $x^n-a^n$ by $x-a$ gives us $$x^n-1 + ax^n-2 +cdots + a^n-1$$

So, $$x^n-a^n=(x-a)(x^n-1 + ax^n-2 +cdots + a^n-1).$$

Replace $x$ & $a$ with $a$ và $b$.

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Just multiply out the right hand side, you"ll see that all terms except for the left hand side cancel.


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