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If x - 3 x - ...

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If $(x - 3) (x - 2)$ is the HCF of the polynomials $x^{2} - 2 x - 3) (2 x^{2} + a x - 2)$ and $(x^{2} + x - 2) (3 x^{2} + b x - 3)$ , then find the value of $a & b$ .

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Solution

Let,
$P (x) = (x^{2} - 2 x - 3) (2 x^{2} + a x - 2) = (x + 1) (x - 3) (2 x^{2} + a x - 2)$

As, HCF is $(x - 3) (x + 2)$
$x + 2 = 0 \Rightarrow x = - 2$

$x = - 2$ is the zero.
$∴ P (- 2) = 0$
$P (- 2) = (- 2 + 1) (- 2 - 3) 2 \times 4 + a (- 2) - 2$
$(- 1) (- 5) (8 - 2 a - 2) = 0$
$5 (6 - 2 a) = 0$
$a = 3$

Let,
$Q (x) = (x^{2} + x - 2) (3 x^{2} + b x - 3) = (x - 1) (x + 2) (3 x^{2} + b x - 3)$

As, HCF is $(x - 3) (x + 2)$
Similarly $Q (3) = 0$

$Q (3) = (3 - 1) (3 + 2) (3 \times 9 + 3 b - 3) (2) (5) (27 + 3 b - 3) = 0 10 (24 + 3 b) = 0 b = - 8$

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