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
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:
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`
`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.