Consider the polynomial function P (x) = − x3 − mx2 + 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

Dan, here's how to do this.

P(x) = -x³ - mx² +nx - 5m

The remainder when P is divided by (x - 2) is P(2) = -(2)³ - m(2²) +n(2) - 5m

We are told that this equals 1 so -8 - 4m + 2n - 5m = 1

=> -9m + 2n = 9 . . . . . . . . . . . . (1)

Similarly, since (x + 1) is a factor, it divides exactly into P => P(-1) = 0

Therefore, -(-1)³ - m(-1)² + n(-1) - 5m = 0

-(-1) - m(1) - n - 5m = 0

=> -6m - n = -1 . . . . . . . . . . . .(2)

To solve these simultaneous equations, add 2 x (2) to (1); -21m = 11

=> m = 7/21 = 1/3

Substitute in (1): -9(1/3) + 2n = 9

-3 + 2n = 9

2n = 12

n = 6

∴ Solution: (m, n) = (1/3, 6)