Question

Consider the polynomial function P(x) = -x^3 - mx^2 + nx - 5m. The remainder when P(x) is divided by (x-2) is 1 and (x+1) is a factor of P(x). Determine m and n algebraically.

Answer 1

P(2) = -8-4m+2n-5m = -9m+2n-8
Since (x+1) is a factor, P(-1) = 0, so 1-m-n-5m=0
So now just solve
-9m+2n-8 = 1
6m+n = 1
That makes P(x) = -x^3 + 1/3 x^2 + 3x + 5/3
= -1/3 (x+1)(3x^2-4x-5)
so clearly, P(-1) = 0
and yes, P(2) = 1

Answer 2

What are the values of m and n tho?