Q. 194.0( 11 Votes )

Find the condition that the zeros of the polynomial f(x) = x3 + 3px2 + 3qx + r may be in Arithmetic Progression.

Answer :

Given: f(x) = x3 + 3px2 + 3qx + r

Concept Used:

For a cubic polynomial: ax3 + bx2 + cx + d = 0

Sum of roots

The product of roots taken two at a time

The product of roots


Let the roots be,

α = a – d

β = a

γ = a + d

Sum of roots = (a – d) + a + (a + d) = 3a


Sum of roots


3a = –3p

a = –p

Since a is the zero of the polynomial, therefore f(a) = 0

f(a) = a3 + 3pa2 + 3qa + r = 0

a3 + 3pa2 + 3qa + r = 0

Substitute a = –p

–p3 + 3p3 – 3pq + r = 0

2p3 – 3pq + r = 0

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Champ Quiz |Revealing the relation Between Zero and Coefficients38 mins
Interactive Quiz - Geometrical Meaning of the Zeroes32 mins
Relationship between Zeroes and Coefficients-238 mins
Relation Between zeroes and Coefficients46 mins
Relationship between Zeroes and Coefficients-152 mins
Quiz - Division Algorithm38 mins
Relation b/w The Zeroes and Coefficients of Cubic Polynomials54 mins
Revision of Relation Between the Zeroes and Coefficients of Quadratic Polynomial46 mins
Interactive Quiz:Polynomials43 mins
Division Algorithm-130 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses