Question

Find the following product:

(2x – y + 3z)(4x² + y² + 9z² + 2xy + 3yz – 6xz)

Answer 1

(2x – y + 3z)(4x² + y² + 9z² + 2xy + 3yz – 6xz)

= 2x(4x² + y² + 9z² + 2xy + 3yz – 6xz) – y(4x² + y² + 9z² + 2xy + 3yz – 6xz) + 3z(4x² + y² + 9z² + 2xy + 3yz – 6xz)

= 8x³ + 2xy² + 18xz² + 4x²y + 6xyz – 12x²z – 4x²y – y³ – 9yz² – 2xy² – 3y²z + 6xyz + 12x²z + 3y²z + 27z³ + 6xyz  + 9yz² – 18xz²

= 8x³ + (2xy² – 2xy²) + (18xz² – 18xz²) + (4x²y – 4x²y) + (6xyz + 6xyz + 6xyz) + (–12x²z + 12x²z) – y³ + (–9yz² + 9yz²) + (–3y²z + 3y²z) + 27z³

= 8x³ + 18xyz – y³ + 27z³

= 8x³ – y³ + 27z³ + 18xyz

Answer 2

(2x – y + 3z)(4x² + y² + 9z² + 2xy + 3yz – 6xz)

= (2x – y + 3z)[(2x)² + (–y)² + (3z)² – (2x)(–y) – (–y)(3z) – (2x)(3z)] 

= (2x)³ + (–y)³ + (3z)³ – 3(2x)(–y)(3z)   ...[Using identity, (a + b + c)(a² + b² + c² – ab – bc – ca) = a³ + b³ + c³ – 3abc]

= 8x³ – y³ + 27z³ + 18xyz