find the remainder when f(x) is divided by (x-2).

Question

The polynomial f(x)=x^4-2x^3+3x^2-ax+b when divided by (x-1) and (x+1) leaves the remainders 5 and 19 respectively.Find the values of a and b.Hence,find the remainder when f(x) is divided by (x-2).

Answer 1

f(x)=x⁴-2x³+3x²-ax+b.

Use synthetic division to divide by zeroes 1 and -1:

1 | 1 -2  3 -a        b

     1  1 -1  2   | 2-a    

     1 -1  2 2-a | 2-a+b (remainder) = 5

-1 | 1 -2 3     -a      b

      1 -1 3  -6   | 6+a    

      1 -3 6 -6-a | 6+a+b (remainder) = 19

So we have 2-a+b=5,  -a+b=3,

                   6+a+b=19, a+b=13.

Therefore (adding these two equations): 2b=16, b=8 and a=13-b=13-8=5.

f(x)=x⁴-2x³+3x²-5x+8.

Remainder after dividing by x-2:

2 | 1 -2 3 -5   8

     1  2 0  6 | 2  

     1  0 3  1 | 10 (remainder). So remainder=10.