When x^4 + a(x^3) + bx +c is divided by (x–1), (x+1), and (x+2), the remainders are 14, 0, and –16 respectively. Find the values of a, b, and c.

Answer 1

This is a bit tricky but use long division to divide the various binomials into the polynomial. Don't forget to include a "0x²" term. When you do, you will get a remainder in terms of a, b, and c and you can set each remainder equal to the remainders given. That will give you 3 equations in 3 unknowns which you can solve by using a variety of techniques. I perfer elimination.

The 3 equations I got were:

1) (from dividing by (x - 1)): a + b + c = 13

2) (from dividing by (x + 1)): a + b - c = 1

3) (from dividing by (x + 2)): 8a + 2b - c = 32

Solving these, I got a = 4, b = 3, and c = 6.

Plugging these into the polynomial and then doing synthetic division for each of the binomials results in the correct remainders so I know I got it correct.