Question

find the zeroes of the polynomial f(x)=x3-21x2+143k-315,if it is given that the zeroes are in arithmetic progression.

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Answer 1

Given:

A polynomial f(x)=x³-21x²+143x-315.

Its roots are in arithmetic progression.

To Find:

The zeroes of the polynomial f(x).

Solution:

Let one of the roots be a .

• Then other roots are:
• a -  d  and a + d ,
• where d is the common difference in Arithmetic Progression.

We know for a cubic equation,

• x³ + bx² + cx + e = 0 ,
• Sum of roots = -b

• Product of the roots = e

From the given f(x) ,

• Sum of roots = a-d + a + a+d = 3a = 21
• Then, a = 7.
• Product of roots = (a-d)a(a+d) = a(a²-d²) = 315
• a² - d² = 315/7 = 45
• d² = 49 - 45
• d² = 4
• d = 2

Therefore roots are 5 ,  7 and 9.

The zeroes of the polynomial f(x)=x3-21x2+143k-315 is 5 , 7 and 9.