2014/033) f(x)=x^4−29x^3+mx^2+nx+k, and f(5)=11, f(11)=17 and f(17)=23, find the sum of m+n+k.

we have f(x) = x+ 6 for x= 5 11 and 17

so f(x)= (x-5)(x-11)(x-17)Q(x) +x + 6
ax f(x) is a 4th oder polynomal so Q(x) = linear say x+ a
coefficient of x3=−29=−5−11−17+a so a = 4

so f(x) = (x-5)(x-11)(x-17)(x+4) + x + 6
put x = 1 to get
1- 29 + m + n + k = (-4) *(-10) *(-16) * 5 + 1 + 6 = - 3200 +7 = - 3193

or m + n + k = -3193 + 28 = - 3165