Let p(x)=x2–5x+a and q(x)=x2–3x+b, where a and b are positive integers. Suppose hof(p(x),q(x)) = x – 1 and k(x) = 1cm (p(x), q(x)). If the coefficient of the highest degree term of k(x) is 1, the sum of the roots of (x – 1) + k(x) is.
7
∴HCF=x−1
⇒p(x)=x2−5x+a
=x2−5x+4
=(x−1)(x−4) .......... (1)
and q(x)=x2−3x+b=x2−3x+2
=(x−1)(x−2) ..........(2)
⇒k(x)=(x−1)(x−2)(x−4)
Hence (x−1)+k(x)=(x−1)+(x−1)(x−2)(x−4)
=(x−1)(x−3)2
Hence sum of roots = 7