2014/048) The polynomial g(x) is cubic. What is the largest value of a if f1(x)=x^2+(a−29)x−a and f2(x)=2x^2+(2a−43)x+a are both factors of g(x)?

f1(x) and f2(x) must have a common factor. Otherwise g(x) shall be product of  f1(x) and f2(x) and order 4

let f1(x) = (x-m)(x-p)

and let f2(x) = 2(x-m)(x-q)

comparing constant term

of  f1(x) = mp = - a and 2 mq = a we get p = - 2q or m = 0 => a = 0

then taking the product and comparing coefficient of x

we get m+p = 29-a ...(1)

m – q = (43-2a)/2

or 2m – p = 43 – 2a ... (2)

solving (1) and (2) 3 m = (72-3a) or m = 24 – a

so p = 2m + 2a – 43 = 48 – 43 = 5

now  - a = mp = 5(24-a) or 4a = 120 or a= 30