If x3+ax2-bx+10 is divisible by x2-3x+2, find the values of a and b.
Let f (x) = x3+ax2-bx+10 and g (x) = x2-3x+2 be the given polynomials.
We have g (x) = x2-3x+2 = (x – 2) (x – 1)
Clearly, (x -1) and (x – 2) are factors of g (x)
Given that f (x) is divisible by g (x)
g (x) is a factor of f (x)
(x – 2) and (x – 1) are factors of f (x)
From factor theorem,
If (x – 1) and (x – 2) are factors of f (x) then f (1) = 0 and f (2) = 0 respectively.
f (1) = 0
(1)3 + a (1)2 – b (1) + 10 = 0
1 + a – b + 10 = 0
a – b + 11 = 0 (i)
f (2) = 0
(2)3 + a (2)2 - b (2) + 10 = 0
8 + 4a – 2b + 10 = 0
4a – 2b + 18 = 0
2 (2a – b + 9) = 0
2a – b + 9 = 0 (ii)
Subtract (i) from (ii), we get
2a – b + 9 – (a – b + 11) = 0
2a – b + 9 – a + b – 11 = 0
a – 2 = 0
a = 2
Putting value of a in (i), we get
2 – b + 11 = 0
b = 13
Hence, a = 2 and b = 13