(c) X^(3)+6x^(2)y+12xy^(2)+7y^(3)

Jasonbradley

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Jun 16, 2024

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Factoring the Cubic Expression: x³ + 6x²y + 12xy² + 7y³

This article explores the factorization of the cubic expression x³ + 6x²y + 12xy² + 7y³. We will use a combination of pattern recognition and the sum of cubes formula to achieve this.

Recognizing a Pattern

First, we notice that the expression exhibits a specific pattern:

• Descending powers of x: The powers of x decrease from 3 to 0.
• Ascending powers of y: The powers of y increase from 0 to 3.
• Coefficients: The coefficients (1, 6, 12, 7) seem to follow a pattern, but it's not immediately clear what it is.

This pattern suggests that the expression might be a perfect cube, but we need to investigate further.

Applying the Sum of Cubes Formula

The sum of cubes formula states:

a³ + b³ = (a + b)(a² - ab + b²)

To apply this formula, we need to find expressions for 'a' and 'b' that satisfy the given cubic expression.

Let's try:

• a = x: x³ matches the first term of the expression.
• b = 2y: (2y)³ = 8y³, which is not directly present in our expression.

Since the coefficient of the y³ term is 7, not 8, we need to adjust our approach slightly. We can rewrite the expression as:

x³ + 6x²y + 12xy² + 7y³ = (x³ + 6x²y + 12xy² + 8y³) - y³

Now, we can apply the sum of cubes formula to the first part of the expression:

x³ + 6x²y + 12xy² + 8y³ = (x + 2y)(x² - 2xy + 4y²)

Therefore, the complete factorization is:

(x + 2y)(x² - 2xy + 4y²) - y³

While we've factored the expression, we can't further simplify the remaining terms using the sum of cubes formula. The final factored form is:

(x + 2y)(x² - 2xy + 4y²) - y³

Conclusion

By recognizing patterns and applying the sum of cubes formula, we were able to factor the cubic expression x³ + 6x²y + 12xy² + 7y³. This process demonstrates how understanding mathematical properties and formulas can be applied to solve complex algebraic problems.