What are the factors of the expression 1331x^3 + 8y^3, and how can we determine them?

To factor the expression 1331x³ + 8y³, we can recognize that this is a sum of cubes. The general formula to factor a sum of cubes a³ + b³ is:

• (a + b)(a² – ab + b²)

In this case, we can identify:

• a = (11x) (since 1331 = 11³)
• b = (2y) (since 8 = 2³)

Applying the sum of cubes formula:

• First, we compute:
• a + b = 11x + 2y
• a² = (11x)² = 121x²
• b² = (2y)² = 4y²
• ab = (11x)(2y) = 22xy

Putting it all together:

• The first factor becomes (11x + 2y)
• The second factor simplifies to:
• 121x² – 22xy + 4y²

Thus, the fully factored form of the expression 1331x³ + 8y³ is:

(11x + 2y)(121x² – 22xy + 4y²)

To summarize, identifying sums of cubes in algebraic expressions like 1331x³ + 8y³ allows us to factor them effectively using known algebraic formulas, leading to a clearer understanding of their structure and roots.