If x = alpha is a positive root of the equati | vedclass

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(A) Let $f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x$.We are given that $f(0) = 0$ and $f(\alpha) = 0$.Since $f(x)$ is a polynomial,it is continuous

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Less than $\alpha$

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(A) Let $f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x$.We are given that $f(0) = 0$ and $f(\alpha) = 0$.Since $f(x)$ is a polynomial,it is continuous on $[0, \alpha]$ and differentiable on $(0, \alpha)$.By Rolle's Theorem,there exists at least one point $c \in (0, \alpha)$ such that $f'(c) = 0$.Note that $f'(x) = na_nx^{n-1} + (n-1)a_{n-1}x^{n-2} + \dots + a_1$.Therefore,the equation $f'(x) = 0$ has at least one positive root in the interval $(0, \alpha)$,which means the root is less than $\alpha$.

If $x = \alpha$ is a positive root of the equation $a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x = 0$,then what can be said about the positive root of the equation $na_nx^{n-1} + (n-1)a_{n-1}x^{n-2} + \dots + a_1 = 0$?

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