Factor Theorem-6

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If x3 + ax2 - x + b has (x - 2) as a factor and leaves a remainder 3 when divided by (x - 3), find a and b.

Solution: Let f(x) = x3 + ax2 - x + b

Since (x - 2) is a factor of f(x), therefore,

f(2) = 23 + a(2)2 - 2 + b = 0

=> 4a + b = - 6 ---------------------------- (i)

Since f(x) is divided by (x - 3), leaves a remainder 3, therefore

f(3) = 3; i.e. 33 + a(3)2 - 3 + b = 3

=> 9a + b = - 21 --------------------------- (ii)

Subtracting eqn (2) from eqn (1), we get

- 5a = 15

=> a = -3

Putting a = -3 in (1), we get 4 (- 3) + b = -6

=> b = 6

Hence, a = -3, b = 6.

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