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Question

The polynomial p(x) = x4 − 2x3 + 3x2 − 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 p(x) is divided by (x − 2).

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Solution

Let:px=x4-2x3+3x2-ax+b

Now,When px is divided by x-1, the remainder is p1.When px is divided by x+1, the remainder is p-1.
Thus, we have:
p1=14-2×13+3×12-a×1+b =1-2+3-a+b =2-a+b
And,
p-1=-14-2×-13+3×-12-a×-1+b =1+2+3+a+b =6+a+b

Now,

2-a+b=5 ...16+a+b=19 ...2

Adding (1) and (2), we get:8+2b=24
2b=16b=8
By putting the value of b, we get the value of a, i.e., 5.
∴ a = 5 and b = 8
Now,
fx = x4-2x3 + 3x2 - 5x + 8
Also,
When px is divided by x-2, the remainder is p2.
Thus, we have:
p2=24-2×23+3×22-5×2+8 a=5 and b=8 =16-16+12-10+8 =10

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