F.4 more about polynomails(2)

a)x^7 + (x^6)y - xy^6 - y^7 = (x^6)x + (x^6)y - xy^6 - y(y^6)= (x^6)(x + y) - (y^6)(x + y)= (x + y) (x^6 - y^6)= (x + y) (x³ - y³) (x³ + y³)= (x + y) (x - y)(x² + xy + y²) (x + y)(x² - xy + y²)= (x - y) (x + y)² (x² - xy + y²) (x² + xy + y²)
b)Let P(x) be the polynomial :(x^6 - y^6) * P(x) = H.C.F. * L.C.M. = (x² - y²) (x^7 + (x^6)y - xy^6 - y^7)
(x + y)² (x² - xy + y²) (x² + xy + y²) * P(x)= (x² - y²) (x - y) (x + y)² (x² - xy + y²) (x² + xy + y²)
P(x)= (x² - y²) (x - y) = x³ - xy² - yx² + y³

2011-04-02 13:03:29 補充：
b) Corrections :

Let P(x) be the polynomial :
(x^6 - y^6) * P(x) = H.C.F. * L.C.M. = (x² - y²) (x^7 + (x^6)y - xy^6 - y^7)


(x - y)(x² + xy + y²) (x + y)(x² - xy + y²) * P(x)

= (x² - y²) (x - y) (x + y)² (x² - xy + y²) (x² + xy + y²)

P(x)

= (x² - y²) (x + y)

= x³ - xy² + yx² - y³