Roots of polynomialProve that the roots of derivative of polynomial of fourth order have the property of arithmetic mean if the roots of polynomial have the same property

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Use the property of the arithmetic mean to find the roots of the polynomial of fourth order:

`x_1=a, x_2=a+d, x_3=a+2d, x_4=a+3d.`

The polynomial is `(x-x_1)(x-x_2)(x-x_3)(x-x_4)=0`

Use the rule of product to find the derivative of polynomial:

`(x-x_2)(x-x_3)(x-x_4)+(x-x_1)(x-x_3)(x-x_4)+(x-x_1)(x-x_2)(x-x_4)+(x-x_1)(x-x_2)(x-x_3)=0`

Divide the equation of derivative by the equation of polynomial.

`1/(x-x_1) + 1/(x-x_2) + 1/(x-x_3) + 1/(x-x_4) = 0`

Substitute the roots.

`1/(x-a) + 1/(x-a-d) + 1/(x-a-2d) + 1/(x-a-3d) = 0`

Put x-a=y

`1/y + 1/(y-3d) + 1/(y-d) + 1/(y-2d) = 0` `(2y-3d)(y^2-3dy+d^2)=0`

`` `2y-3d = 0 =gt y_2=3d/2`

`` `y_1+y_3=3d =gt y_1+y_3=2y_2`

The roots of the derivative of polynomial are:`x'_1=y_1+a; x'_2=y_2+a; x'_3=y_3+a`

ANSWER: Since between these roots there is the relation `x'_1+x'_3=2x'_2` , it means that the roots of derivative are the elements of an arithmetic progression.

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