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Question

Find the values of a and b so that the polynomial (x310x2+ax+b) is  exactly divisible by (x-1) as well as (x-2).


Solution

Let the given polynomial be
f(x)=x3+10x2+ax+b
If f(x) is exactly divisible by (x - 1).
f(1) = 0
13+10×12+a×1+b=0
1+10+a+b=0
a+b=11(1)
If f(x) is exactly divisible by (x - 2)
f (2) = 0
23+10×22+a×2+b=0
8+40+2a+b=0
2a+b=48(2)
Subtracting eq (2) by eq (1):
- a = - 11 + 48
- a = 37
a=37
Put a = - 37 in eq (1):
- 37 + b = - 11
b = - 11 + 37
b = 26

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