Question

Multiply: x²+4y²+z²+2xy+xz-2yz by x-2y-z

Answer 1

(x2+4y2+z2+2xy+xz–2yz)(−z+x–2y)
=x2(−z+x–2y)+4y2(−z+x–2y)+z2(−z+x−2y)+2xy(−z+x−2y)+xz(−z+x−2y)−2yz(−z+x−2y)
=−x2z+x3–2x2y–4y2z+4xy2–8y3–z3+xz2−2yz2−2xyz+2x2y−4xy2−xz2+x2z−2xyz+2yz2−2xyz+4y2z
=(−x2z+x2z)+x3+(−2x2y+2x2y)+(−4y2z+4y2z)+(4xy2−4xy2)−8y3−z3+(xz2−xz2)+(−2yz2+2yz2)+(−2xyz−2xyz−2xyz)
=x3−8y3−z3−6xyz

Alternate method:

(x–2y–z)(x2+4y2+z2+2xy+xz−2yz)
=(x−2y−z)[(x)2+(−2y)2+(−z)2−(x)(−2y)−(−2y)(−z)−(x)(−z)]
=(x)3+(−2y)3+(−z)3–3(x)(−2y)(−z)
[Usingtheidentity,a3+b3+c3−3abc=(a+b+c)(a2+b2+c2–ab–bc–ca)]
=x3–8y3–z3–6xyz