Question

Given the LCM and GCD of the two polynomials p(x) and q(x) find the unknown polynomial in the following table

LCM	GCD	p(x)	q(x)
(x4 –  y4)(x4 + x2y2 + y2)	(x2 –  y2)		(x4 –  y4)(x2 + y2 –  xy)

Answer 1

L.C.M. (x² + y²)(x⁴ + x²y² + y⁴)

(x² + y²)[(x² + y²)²-(xy)²]

(x² + y²) (x² + y² + xy) (x² + y² – xy)

G.C.D. = x² – y²

(x + y)(x – y)

q(x) = (x⁴ – y⁴) (x² + y² – xy)

= [(x²)² – (y²)²](x² + y² – xy)

= (x² + y²) (x² – y²) (x² + y² – xy)

(x² + y²) (x + y) (x – y) (x² + y² – xy)

P(x) = x² + y² + xy

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

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

p(x) = x² + y² + xy