A structural engineer models the stress L(x)L(x)L(x) on a cantilever beam at a distance x x\,x from the support as
L(x)=x3+(k−3)x2−2x+m L(x) = x^3 + (k - 3)x^2 - 2x + m L(x)=x3+(k−3)x2−2x+mwhere k k\,k and m m\,m are constants and k>0k > 0k>0.
Given that (x−4)(x - 4)(x−4) is a factor of L(x)L(x)L(x):
Show that 16k+m=−816k + m = -816k+m=−8.
Given also that when L(x)L(x)L(x) is divided by (x+k)(x + k)(x+k), the remainder is -48:
Show that 3k2−2k−m−48=03k^2 - 2k - m - 48 = 03k2−2k−m−48=0.
Hence find the value of k k\,k and the value of mmm.
Find a quadratic expression g(x)g(x)g(x) such that L(x)=(x−4)g(x)L(x) = (x - 4)g(x)L(x)=(x−4)g(x).
