G.C.D. (H.C.F.) AND L.C.M. OF POLYNOMIALS
The following formulae will be helpful in factorizing polynomials.
(i) \( (x+y)^2 = x^2+ 2xy + y^2 \)
(ii) \( (x-y)^2 = x^2-2xy+y^2 \)
(iii) \( (x+y+z)^2 = x^2 + y^2 +z^2+2xy+2yz+2zx \)
(iv) \( (x+y)^3 = x^3+3xy(x+y) +y^3 \)
(v) \( (x-y)^3 = x^3-3xy(x-y) -y^3 \)
(vi) \( x^2-y^2 = (x-y) (x+y) \)
(vii) \( x^3+y^3 = (x+y) (x^2-xy+y^2) \)
(viii) \( x^3-y^3 = (x-y) (x^2+xy+y^2) \)
(ix) \( x^3+y^3+z^3-3xyz = (x+y+z) (x^2+y^2+z^2-xy-yz-zx) \)
(x) \( x^4-y^4 = (x^2-y^2) (x^2+y^2) = (x-y) (x+y) (x^2+y^2) \)
(xi)\( x^8-y^8 = (x^4-y^4) (x^4+y^4) = (x-y) (x+y) (x^2+y^2) (x^4+y^4) \)
(xii) \( x^4+y^4+x^2y^2 = (x^4+y^4+sx^2y^2) – x^2y^2 = (x^2+y^2)2 -(xy)^2 = (x^2+y^2+xy) (x^2+y^2-xy) \)