Question

If H(x) and L(x) denote the HCF and LCM of two polynomials f(x) and g(x) such that

H(x) + L(x) = f(x) + g(x) then prove that [H(x)]3 + [L(x)]3 = [f(x)]

3 + [g(x)]3




Solve step by step please help
Right And easy answer please ​

Answer 1

Answer:

Given GCD =x+1 and LCM =x

6

−1

Let f(x)=x

3

+1.

We known that LCM×GCD=f(x)×g(x)

→g(x)=

f(x)

LCM×GCD

=

x

3

+1

(x

6

−1)(x+1)

x

3

+1

(x

3

+1)(x

3

−1)(x+1)

=(x

3

−1)(x+1)

Hence, g(x)=(x

3

−1)(x+1).