Question

Simplify the following:

(2x + 5y)³ – (2x – 5y)³

Answer 1

Step 1: Expand the first term

We expand the first term (2x + 5y)³ using the binomial formula (a + b)³ = a³ + 3a²b + 3ab² + b³:

(2x + 5y)³ = (2x)³ + 3(2x)²(5y) + 3(2x)(5y)² + (5y)³

(2x + 5y)³ = 8x³ + 3(4x²)(5y) + 3(2x)(25y²) + 125y³

(2x + 5y)³ = 8x³ + 60x²y + 150xy² + 125y³

Step 2: Expand the second term

We expand the second term (2x – 5y)³ using the binomial formula (a – b)³ = a³ – 3a²b + 3ab² – b³:

(2x – 5y)³ = (2x)³ – 3(2x)²(5y) + 3(2x)(5y)² – (5y)³

(2x – 5y)³ = 8x³ – 3(4x²)(5y) + 3(2x)(25y²) – 125y³

(2x – 5y)³ = 8x³ – 60x²y + 150xy² – 125y³

Step 3: Subtract the expanded terms

Subtract the expanded second term from the first term:

(2x + 5y)³ – (2x – 5y)³

= (8x³ + 60x²y + 150xy² + 125y³) – (8x³ – 60x²y + 150xy² – 125y³)

= 8x³ + 60x²y + 150xy² + 125y³ – 8x³ + 60x²y – 150xy² + 125y³

Step 4: Combine like terms

Group and combine the like terms:

= (8x³ – 8x³) + (60x²y + 60x²y) + (150xy² – 150xy²) + (125y³ + 125y³)

= 0 + 120x²y + 0 + 250y³

= 120x²y + 250y³