Question

using euclid division algorithm prove that if x + a is a factor of x cube + b x square + cx + d then minus a 4 + a square b minus a c + d equal to zero

Answer 1

Is x^2+px+1 is a factor of ax^3+bx+c, therefore factor theorem states that the remainder when ax^3+bx+c is divided by x^2+px+1 will be zero.

Thus,

(b-a)x+c-apx^2=0

i.e (b-a)x+c-apx^2=0\c

x+0+0\c x^2

Comparing the coefficients we have,

b-a=0 (1)

c=0 (2)

ap=0 (3)

We need to prove,

a^2-c^2=ab

i.e a^2-c^2-ab=0

LHS

=a^2-0-a

using (2)

=a(a-b)

=a\c

using (1)

=0

=RHS

Hence Proved.