Question

Find the product: 

(3a + b + 5c) (9a² + b² + 25c² – 3ab – 5bc – 15ас)

Answer 1

Given expression: (3a + b + 5c)(9a² + b² + 25c² – 3ab – 5bc – 15ас)

Step-wise calculation:

Recognize this form as the product of a sum of three terms and a specific trinomial expression.

This expression matches the identity for the expansion of the cube of a trinomial:

(x + y + z)(x² + y² + z² – xy – yz – zx) 

= x³ + y³ + z³ – 3xyz

Identify:

• x = 3a
• y = b
• z = 5c

Substitute into the identity:

(3a)³ + b³ + (5c)³ – 3 × 3a × b × 5c

Calculate each term:

• (3a)³ = 27a³
• b³ = b³
• (5c)³ = 125c³
• 3 × 3a × b × 5c = 45abc

So, the product is 27a³ + b³ + 125c³ – 45abc.