Factor by any method. See Examples 1–7. $1000x^3+343y^3$

Verified step by step guidance

1. Recognize that the expression \$1000x^3 + 343y^3\( is a sum of cubes because \)1000x^3 = (10x)^3\( and \)343y^3 = (7y)^3$.
2. Recall the sum of cubes factoring formula: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
3. Identify \(a = 10x\) and \(b = 7y\) in the given expression.
4. Apply the formula by substituting \(a\) and \(b\): write the factorization as \((10x + 7y)((10x)^2 - (10x)(7y) + (7y)^2)\).
5. Simplify the terms inside the second factor: calculate \((10x)^2\), \((10x)(7y)\), and \((7y)^2\) to complete the factorization.

Key Concepts

Sum of Cubes Formula

The sum of cubes formula states that a³ + b³ = (a + b)(a² - ab + b²). This formula is used to factor expressions where two terms are each perfect cubes added together. Recognizing 1000x³ and 343y³ as cubes allows applying this formula directly.

Identifying Perfect Cubes

A perfect cube is a number or expression raised to the third power, such as 1000 = 10³ and 343 = 7³. Identifying each term as a perfect cube is essential before applying the sum of cubes formula, ensuring the expression fits the pattern a³ + b³.

Factoring Polynomials

Factoring polynomials involves rewriting an expression as a product of simpler polynomials. Using special formulas like the sum of cubes helps break down complex expressions into factors, which can simplify solving equations or further algebraic manipulation.