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.
2. Recall the formula for factoring a sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
3. Identify \(a\) and \(b\) in the expression: \(a = 10x\) and \(b = 7y\), since \((10x)^3 = 1000x^3\) and \((7y)^3 = 343y^3\).
4. Substitute \(a = 10x\) and \(b = 7y\) into the sum of cubes formula: \((10x + 7y)((10x)^2 - (10x)(7y) + (7y)^2)\).
5. Simplify the expression: \((10x + 7y)(100x^2 - 70xy + 49y^2)\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This process is essential for simplifying expressions and solving equations. Common methods include factoring out the greatest common factor, using special products like the difference of squares, and applying the sum or difference of cubes formulas.

Sum and Difference of Cubes

The sum and difference of cubes are specific algebraic identities used to factor expressions of the form a^3 + b^3 and a^3 - b^3. The formulas are a^3 + b^3 = (a + b)(a^2 - ab + b^2) and a^3 - b^3 = (a - b)(a^2 + ab + b^2). Recognizing these forms is crucial for efficiently factoring cubic polynomials.

Polynomial Degree and Terms

The degree of a polynomial is the highest power of the variable in the expression, which determines its behavior and the number of roots it can have. In the expression 1000x^3 + 343y^3, the degree is 3, indicating it is a cubic polynomial. Understanding the structure of polynomials helps in applying appropriate factoring techniques.