First slide

Theory of equations

Question

$If f (x) = (x^{2} + 3 x + 2) (x^{2} - 7 x + a) and g (x) = (x^{2} - x - 12) (x^{2} + 5 x + b), then the values of a and b, if (x + 1) (x - 4) is HCF of f (x) and g (x), are$

Moderate

A

$a = 10; b = 6$
B

$a = 4; b = 12$
C

$a = 12; b = 4$
D

$a = 6; b = 10$

Solution

$f (x) = (x^{2} + 3 x + 2) (x^{2} - 7 x + a) = (x + 1) (x + 2) (x^{2} - 7 x + a)$
f(x) is divisible by (x+1)(x-4)
$x^{2} - 7 x + a is divisible by x - 4$ $\Rightarrow a = 12$
similarly $g (x) = (x^{2} - x - 12) (x^{2} + 5 x + b)$
= $(x - 4) (x + 3) (x^{2} + 5 x + b)$
$x^{2} + 5 x + b is divisible by x + 1$
$\Rightarrow b = 4$

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