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## Homework Statement

Prove that if f(a) = 0, then f(x) = (x-a)g(x) where f and g are polynomial functions.

## Homework Equations

[tex]

x^n-a^n = (x-a)h_n(x)

[/tex]

where h

_{n}(x) is a polynomial function.

## The Attempt at a Solution

[tex]

\begin{align*}

f(x) &= f(x) - f(a)\\

f(x) &= [\lambda_n x^n + \lambda_{n-1} x^{n-1} + ... + \lambda_1 x + \lambda_0] - [\lambda_n a^n + \lambda_{n-1} a^{n-1} + ... + \lambda_1 a + \lambda_0]\\

f(x) &= \lambda_n (x^n-a^n) + \lambda_{n-1} (x^{n-1} - a^{n-1}) +...+ \lambda_1(x-a)+(\lambda_0 - \lambda_0)\\

f(x) &= \lambda_n (x-a)h_n(x) + \lambda_{n-1} (x - a)h_{n-1}(x) +...+ \lambda_1(x-a)\\

f(x) &= (x-a)g(x)

\end{align*}

where $ g(x) = \lambda_nh_n(x) + \lambda_{n-1}h_{n-1}(x) +...+ \lambda_1$

[/tex]

Now, I think what I've done here is valid. But I assumed that the relevant equation that I posted is true, which (I think) it is. Can someone tell me if this is an acceptable proof?

Thanks.

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