Question

let p(x) be a cubic polynomial with leading coefficient 1. if p(1)=2p(2)=3p(3)=1 and s be the sum of roots of p(x), then the value of 3s

Answer 1

Assume the cubic polynomial p(x)=x3+ax2+bx+cp(x)=x3+ax2+bx+c

p(1)=1+a+b+c=1=>a+b+c=0=>c=−a−b—(1)p(1)=1+a+b+c=1=>a+b+c=0=>c=−a−b—(1)

2∗p(2)=1=>p(2)=12=8+4a+2b+c=8+4a+2b−a−b2∗p(2)=1=>p(2)=12=8+4a+2b+c=8+4a+2b−a−b

=>3a+b=12−8=−152=>b=−3a−152—(2)=>3a+b=12−8=−152=>b=−3a−152—(2)

3∗p(3)=1=>p(3)=13=33+a∗32+b∗3+c—(3)3∗p(3)=1=>p(3)=13=33+a∗32+b∗3+c—(3)

Substitute the values of b and c from (1) (2) in (3)

=>13=27+9a+3∗(−3a−152)+(−a−b)