A³ - b³

Definition

The expression a³ - b³ represents the difference between the cubes of two variables, a and b. This term is particularly relevant in the context of factoring special products, as it is one of the common factorization patterns that students encounter in algebra.

5 Must Know Facts For Your Next Test

1. The expression a³ - b³ can be factored using the difference of cubes formula: a³ - b³ = (a - b)(a² + ab + b²).
2. Factoring a³ - b³ is useful in simplifying algebraic expressions and solving equations.
3. The difference of cubes formula is one of the special product formulas that students must be familiar with in order to factor polynomials effectively.
4. The factors of a³ - b³ are (a - b) and (a² + ab + b²), which can be used to solve a variety of algebraic problems.
5. Understanding the difference of cubes formula and its application is crucial for success in factoring and solving more complex polynomial expressions.

Review Questions

• Explain the difference of cubes formula and how it relates to the expression a³ - b³.
  • The difference of cubes formula states that a³ - b³ = (a - b)(a² + ab + b²). This formula allows you to factor the expression a³ - b³ into the product of two binomials: (a - b) and (a² + ab + b²). Understanding this formula is essential for factoring polynomial expressions that involve the difference of two cubes, as it provides a systematic approach to breaking down the expression into its constituent factors.
• Describe the steps involved in factoring the expression a³ - b³ using the difference of cubes formula.
  • To factor the expression a³ - b³ using the difference of cubes formula, follow these steps: 1. Identify the variables a and b in the expression. 2. Apply the difference of cubes formula: a³ - b³ = (a - b)(a² + ab + b²). 3. The first factor is (a - b), and the second factor is (a² + ab + b²). 4. Verify that the product of the two factors equals the original expression, a³ - b³.
• Explain how the factorization of a³ - b³ can be used to solve algebraic problems.
  • The factorization of a³ - b³ into (a - b)(a² + ab + b²) can be applied to solve a variety of algebraic problems. For example, if you are asked to simplify an expression involving a³ - b³, you can use the difference of cubes formula to rewrite the expression as the product of the two binomial factors. This can lead to further simplification or manipulation of the expression, depending on the context of the problem. Additionally, the factorization can be used to solve equations that contain a³ - b³ by setting one of the factors equal to zero and solving for the variable.

Related terms

Cube: The cube of a number is the result of multiplying that number by itself three times. For example, the cube of 2 is 2 × 2 × 2 = 8.

Difference of Cubes:

The difference of cubes is a special product where the expression is the difference between the cubes of two variables or numbers.

Factoring:

Factoring is the process of breaking down a polynomial expression into a product of simpler expressions, such as factors.