The HCF of the functions
$x^{3} + (a + b) x^{2} + (a b + 1) x + b$ and $x^{3} + 2 a x^{2} + (a^{2} - 1) x + a$ is

x^{2} + a x + 1

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x^{2} + b x + 1

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x^{2} + x + a

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x^{2} + x + b

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Solution

The correct option is A $x^{2} + a x + 1$
x³ + (a +b)x² + (ab + 1)x + b
= x³ + (a +b)x² + abx + x + b
= x [x² + (a + b)x + ab] + (x + b)
= x (x + a) (x + b) + (x + b)
= (x + b) [x (x + a) + 1]
= (x + b) (x² + ax + a)

x ³ + 2ax ² + (a ² + 1)x + a
= x³ + 2ax² + a ² x + x + a
= x (x ² + 2ax + a ² ) + (x + a)
= x (x + a) (x + a) + (x + a)
= (x + a) [x (x + a) + 1]
= (x + a) (x ² + ax + 1)
Common factor between the two polynomials = x ² + ax + 1
∴ HCF = x ² + ax +1

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