Factorization of polynomials

Could someone explain to lớn me how the summation of the the geometric series explains the factorization?  The long parenthesized term is a geometric series with first term \$a^n-1\$ and ratio \$frac ba\$ so mix \$x=frac ba\$ 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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