Question

Simplify the following:

(3x + 5y) (9x² – 15xy + 25y²) – (3x – 5y) (9x² + 15xy + 25y²)

Answer 1

We simplify the expression:

(3x + 5y)(9x² – 15xy + 25y²) – (3x – 5y)(9x² + 15xy + 25y²)

Observe that each term is of the form:

(a + b)(a² – ab + b²) = a³ + b³

(a – b)(a² + ab + b²) = a³ – b³

Here:

• a = 3x
• b = 5y

Step 1: Convert each part using identities

First term:

(3x + 5y)(9x² – 15xy + 25y²) = (3x)³ + (5y)³

(3x + 5y)(9x² – 15xy + 25y²) = 27x³ + 125y³

Second term:

(3x – 5y)(9x² + 15xy + 25y²) = (3x)³ – (5y)³

(3x – 5y)(9x² + 15xy + 25y²) = 27x³ – 125y³

Step 2: Subtract

(27x³ + 125y³) – (27x³ – 125y³)

Distribute the minus:

= 27x³ + 125y³ – 27x³ + 125y³

Combine like terms:

27x³ – 27x³ = 0

125y³ + 125y³ = 250y³