CBSE Class 10 Answered
Find the condition such that the roots of the polynomial are in AP ?
Asked by rushabh1234 | 14 Mar, 2019, 11:13: AM
Expert Answer
SOLUTION :
Let, α = a - d, β = a and γ = a + d be the zeroes of the polynomial.
And let the polynomial be 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
Answered by Arun | 14 Mar, 2019, 09:27: PM
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