If the polynomial equation a_0x^ n +a_ n-1 x^ n-1 +a_ n-2 x^ n-2 +...a_2x^2+a_1x+a_0=0 n positive integer,has two different real roots and , then between and , the equation na_ n x^ n-1 +(n-1)a_ n-1 x^ n-2 +...+a_1=0 has: 1. Exactly one root. 2. Almost one root. 3. At least one root. 4. No root.

3. At least one root. Solution: We observe that, na_ n x^ n-1 +(n-1)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_2x^2+a_1x+a_0=0Polynomial function is continuous everywhere in R and concequently derivative in R. Therefore, a_ n x^ n +a_ n-1 x^ n-1 +a_ n-2 x^ n-2 +...a_2x^2+a_1x+a_0 is continuous on , and derivative on , . Hence, it is satisfies the both the conditions of Rolle's theorem. By algebric interpretation of Roll's theorem, we know that between any two roots of a function f(x), there exists atleast one root of its derivative. Hence, the equation na_ n x^ n-1 +(n-1)a_ n-1 x^ n-2 +...+a_1=0 will have atleast one root between and .

If the polynomial equation a_0x^ n +a_ n-1 x^ n-1 +a_ n-2 x^ n-2 …

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Question

If the polynomial equation $\text{a}_0\text{x}^{\text{n}}+\text{a}_{\text{n}-1}\text{x}^{\text{n}-1}+\text{a}_{\text{n}-2}\text{x}^{\text{n}-2}+...\text{a}_2\text{x}^2+\text{a}_1\text{x}+\text{a}_0=0$ n positive integer,has two different real roots $\alpha$ and $\beta,$ then between $\alpha$ and $\beta,$ the equation $\text{n}\text{a}_{\text{n}}\text{x}^{\text{n}-1}+(\text{n}-1)\text{a}_{\text{n}-1}\text{x}^{\text{n}-2}+...+\text{a}_1=0$ has:

Exactly one root.
Almost one root.
At least one root.
No root.

✓

Answer

At least one root.

Solution:
We observe that, $\text{n}\text{a}_{\text{n}}\text{x}^{\text{n}-1}+(\text{n}-1)\text{a}_{\text{n}-1}\text{x}^{\text{n}-2}+...+\text{a}_1=0$ is the derivative of the polynomial $\text{a}_{\text{n}}\text{x}^{\text{n}}+\text{a}_{\text{n}-1}\text{x}^{\text{n}-1}+\text{a}_{\text{n}-2}\text{x}^{\text{n}-2}+...\text{a}_2\text{x}^2+\text{a}_1\text{x}+\text{a}_0=0$Polynomial function is continuous everywhere in R and concequently derivative in R.
Therefore, $\text{a}_{\text{n}}\text{x}^{\text{n}}+\text{a}_{\text{n}-1}\text{x}^{\text{n}-1}+\text{a}_{\text{n}-2}\text{x}^{\text{n}-2}+...\text{a}_2\text{x}^2+\text{a}_1\text{x}+\text{a}_0$ is continuous on $\alpha,\beta$ and derivative on $\alpha,\beta.$
Hence, it is satisfies the both the conditions of Rolle's theorem.
By algebric interpretation of Roll's theorem, we know that between any two roots of a function f(x), there exists atleast one root of its derivative.
Hence, the equation $\text{n}\text{a}_{\text{n}}\text{x}^{\text{n}-1}+(\text{n}-1)\text{a}_{\text{n}-1}\text{x}^{\text{n}-2}+...+\text{a}_1=0$ will have atleast one root between $\alpha$ and $\beta.$

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