The polynomial -6x^3 + 50x^2 + ax +24, has factors x – 1 and 2x + 1. What is the value of a?
Looking at the lead coefficient, we see that we need
-(x-1)(2x+1)(3x+k) = -6x^3 + (3-2k)x^2 + (3+k)x + k
so, k = 24 and that gives us
-6x^3 - 45x^2 + 27x + 24
Now we have a problem, since we need 3-2k = 50 ...
This is impossible, as 3-2k would be an odd number, while 50 is even. Therefore, there is no value of a that satisfies the given conditions.
or
let f(x) = -6x^3 + 50x^2 + ax +24
we are told that x-1 is a factor, so f(1) = 0
f(1) = -6 + 50 + a + 24 = 0
a = -68
we are also told that 2x + 1 is a factor, so f(-1/2) = 0
f(-1/2) = -6/-8 + 50/4 -a/2 + 24 = 0
≠ -68
So we have a contradiction. So there is no solution.