find the condition that the zeros of the polynomial f(x) = x³+3px²+3qx + r may be in A.P.
Answers
Answered by
82
Let, α = a - d, β = a and γ = a + d be the zeroes of the polynomial.
Given : f(x) = x³ + 3px² + 3qx + r
Sum of zeroes = −coefficient of x² / coefficient of x³
α + β + γ = −b/a
α + β + γ = - 3p/1 = -3p
(a – d) +( a) + (a + d) = -3p
a + a + a -d -d = -3p
3a = -3p
a = -3p × ⅓ = -p
a = -p …………………..(1)
Since, a is the zero of the polynomial f(x),
Therefore, f(a) = 0
f(a)= a³ +3pa² + 3qa + r
a³ +3pa² + 3qa + r = 0
On Substituting a = -p ,
= (−p)³ + 3p(-p)² + 3q(-p) + r=0
= −p³ +3p³ – 3pq + r=0
= 2p³ –3pq + r=0
Hence, the condition for the Given polynomial is 2p³ –3pq + r = 0.
HOPE THIS ANSWER WILL HELP YOU….
Answered by
15
Answer:
Step-by-step explanation:
Attachments:
Similar questions