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# Solution for If the Polynomial Equation N Positive Integer, Has Two Different Real Roots α and β, Then Between α and β, the Equation A) Exactly One Root (B) Almost One Root (C) at Least One Root (D) No Root - CBSE (Commerce) Class 12 - Mathematics

ConceptMaximum and Minimum Values of a Function in a Closed Interval

#### Question

If the polynomial equation $a_0 x^n + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + . . . + a_2 x^2 + a_1 x + a_0 = 0$ n positive integer, has two different real roots α and β, then between α and β, the equation $n \ a_n x^{n - 1} + \left( n - 1 \right) a_{n - 1} x^{n - 2} + . . . + a_1 = 0 \text { has }$.

(a) exactly one root
(b) almost one root
(c) at least one root
(d) no root

#### Solution

(c) at least one root

We observe that, $n a_n x^{n - 1} + \left( n - 1 \right) a_{n - 1} x^{n - 2} + . . . + a_1 = 0$ is the derivative of the polynomial $a_n x^n + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + . . . + a_2 x^2 + a_1 x + a_0 = 0$

Polynomial function is continuous every where in R and consequently derivative in R
Therefore, $a_n x^n + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + . . . + a_2 x^2 + a_1 x + a_0$ is continuous on

$\left[ \alpha, \beta \right]$ and derivative on $\left( \alpha, \beta \right)$.
Hence, it satisfies the both the conditions of Rolle's theorem.
By algebraic interpretation of Rolle's theorem, we know that between any two roots of a function $f\left( x \right)$ , there exists at least one root of its derivative.
Hence, the equation
$n a_n x^{n - 1} + \left( n - 1 \right) a_{n - 1} x^{n - 2} + . . . + a_1 = 0$ will have at least one root between $\alpha \text{ and } \beta$.
Is there an error in this question or solution?

#### APPEARS IN

Solution If the Polynomial Equation N Positive Integer, Has Two Different Real Roots α and β, Then Between α and β, the Equation A) Exactly One Root (B) Almost One Root (C) at Least One Root (D) No Root Concept: Maximum and Minimum Values of a Function in a Closed Interval.
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