Question

The HCF of the functions
x^3 + (a + b)x^2 +(ab + 1)x + b and x^3 + 2ax^2 + (a^2 + 1)x + a is

Answer 1

Answer:

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