If a polynomial equation a_nx^n + a_{n-1}x^{n | vedclass

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(C) Let $f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$. Since $f(x)$ is a polynomial,it is continuous on the closed interval $[\alpha, \beta]$

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At least one root

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(C) Let $f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$. Since $f(x)$ is a polynomial,it is continuous on the closed interval $[\alpha, \beta]$ and differentiable on the open interval $(\alpha, \beta)$. Given that $\alpha$ and $\beta$ are roots of $f(x) = 0$,we have $f(\alpha) = 0$ and $f(\beta) = 0$. According to Rolle's Theorem,there exists at least one point $c \in (\alpha, \beta)$ such that $f'(c) = 0$. The derivative of the polynomial is $f'(x) = na_nx^{n-1} + (n - 1)a_{n-1}x^{n-2} + \dots + a_1$. Therefore,the equation $f'(x) = 0$ has at least one root in the interval $(\alpha, \beta)$.

If a polynomial equation $a_nx^n + a_{n-1}x^{n-1} + \dots + a_2x^2 + a_1x + a_0 = 0$,where $n$ is a positive integer,has two distinct roots $\alpha$ and $\beta$,then how many roots does the equation $na_nx^{n-1} + (n - 1)a_{n-1}x^{n-2} + \dots + a_1 = 0$ have in the interval $(\alpha, \beta)$?

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