When $x^{3} - 2 x^{2} + a x - b$ is divided by $x^{2} - 2 x - 3$ , the remainder is x-6.what are the values of a and b.

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Solution

$\frac{x^{3} - 2 x^{2} + a x - b}{x^{2} - 2 x - 3}; x^{2} - 2 x - 3 = (x - 3) (x + 1) F o r, x = 3; 27 - 18 + 3 a - b = - 3 \Rightarrow 3 a - b = - 12 (1) [∵ x - 6 i s r e m a i n d e r] F o r, x = - 1; - 1 - 2 - a - b = - 7 \Rightarrow a + b = 4 (2) [∵ x - 6 i s r e m a i n d e r] A d d (1) + (2), 3 a + a = - 8 \Rightarrow a = - 2 ∴ b = 6$ .

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