The expression $a^3 - b^3$ represents the difference between the cubes of two variables, $a$ and $b$. This term is particularly important in the context of factoring polynomials, as it can be factored using a specific method known as the difference of cubes.
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The expression $a^3 - b^3$ can be factored using the difference of cubes formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.
Factoring $a^3 - b^3$ is useful in simplifying algebraic expressions, solving polynomial equations, and understanding the properties of polynomials.
The factors of $a^3 - b^3$ are $(a - b)$ and $(a^2 + ab + b^2)$, which are both polynomials.
The difference of cubes formula works because $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, which can be verified by expanding the right-hand side.
Factoring $a^3 - b^3$ is a common technique used in college algebra and is often seen in problems related to polynomial factorization.
Review Questions
Explain the difference of cubes formula and how it is used to factor the expression $a^3 - b^3$.
The difference of cubes formula states that $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. This formula can be used to factor the expression $a^3 - b^3$ by recognizing that it is in the form of a difference of cubes. By applying the formula, the original expression can be factored into the product of two smaller polynomials: $(a - b)$ and $(a^2 + ab + b^2)$. This factorization can simplify the expression and make it easier to work with in various algebraic manipulations and problem-solving contexts.
Describe the relationship between the difference of cubes and perfect cubes, and explain how this relationship is useful in factoring polynomials.
The difference of cubes, $a^3 - b^3$, is related to perfect cubes because it involves the difference between two cubes. A perfect cube is a number or expression that can be written as the cube of another number or expression, such as $8 = 2^3$ or $27 = 3^3$. The difference of cubes formula exploits this relationship by factoring $a^3 - b^3$ into the product of $(a - b)$ and $(a^2 + ab + b^2)$. This factorization is useful in simplifying algebraic expressions, solving polynomial equations, and understanding the properties of polynomials, as it allows complex expressions to be broken down into more manageable factors.
Analyze the importance of the difference of cubes factorization in the broader context of polynomial factorization and its applications in college algebra.
The difference of cubes factorization, $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, is a crucial technique in the broader context of polynomial factorization. It allows for the simplification of complex polynomial expressions by breaking them down into smaller, more manageable factors. This is particularly important in college algebra, where students are often required to solve polynomial equations, simplify algebraic expressions, and understand the properties of polynomials. The ability to recognize and apply the difference of cubes formula demonstrates a deeper understanding of polynomial factorization and equips students with a valuable tool for solving a wide range of algebraic problems. Furthermore, the concepts and skills developed through working with the difference of cubes factorization can be extended to other factorization methods and polynomial manipulations, making it a foundational topic in college algebra.
Related terms
Difference of Cubes: The difference of cubes is a factorization method used to factor polynomial expressions in the form $a^3 - b^3$.
Perfect Cube: A perfect cube is a number or expression that can be written as the cube of another number or expression, such as $8 = 2^3$ or $27 = 3^3$.
Polynomial Factorization: Polynomial factorization is the process of expressing a polynomial as a product of smaller polynomials, which can make solving polynomial equations and expressions easier.