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Find the value of a and b such that $x^{2} - 2 x - 3$ is a factor of the polynomial $x^{3} - 3 x^{2} + ax - b$ .

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-1,-2

-2,-4

-1,-4

-2,-3

answer is C.

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Detailed Solution

x^{2} - 2 x - 3 = (x - 3) (x + 1)

x=3 and x = 1 are the roots of

x^{3} - 3 x^{2} + ax - b = 0

3^{3} - 3 \times 3^{2} + a \times 3 - b = 0

\Rightarrow

b=3a
And,

{(- 1)}^{3} - 3 \times {(- 1)}^{2} + a \times (- 1) - b = 0

\Rightarrow - 1 - 3 - a - b = 0

\Rightarrow a + b = - 4

\Rightarrow a = - 1, b = - 3

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