Q. 184.1( 15 Votes )

# If x3

Answer :

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

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