Question

Find the GCD pair of the following polynomials

(x³ + y³), (x⁴ + x²y² + y⁴) whose LCM is (x³ + y³) (x² + xy + y²)

Answer 1

p(x) = x³ + y³

= (x + y)(x² – xy + y²)

g(x) = x⁴ + x²y² + y⁴ = [x² + y²]² – (xy)²

= (x² + y² + xy) (x² + y² – xy)

L.C.M. = (x³ + y³) (x² + xy + y²)

(x + y) (x² – xy + y²) (x² + xy + y²)

G.C.D. = `("p"(x) xx "g"(x))/("L"."C"."M".)`

= `((x + y)(x^2 - xy + y^2) xx (x^2 + y^2 + xy)(x^2 + y^2 - xy))/((x + y)(x^2 - xy + y^2)(x^2 + xy + y^2))`

G.C.D. = x² – xy + y²