key term - A^3 + b^3

Definition

The expression $a^3 + b^3$ represents the sum of the cubes of two variables, $a$ and $b$. This expression is particularly important in the context of factoring polynomials, as it can be factored using a special factorization method known as the sum of cubes formula.

5 Must Know Facts For Your Next Test

1. The expression $a^3 + b^3$ can be factored using the sum of cubes formula, which states that $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.
2. The sum of cubes formula is particularly useful when dealing with polynomials that have a difference of two cubes, such as $x^3 - y^3$, which can be factored as $(x - y)(x^2 + xy + y^2)$.
3. Factoring $a^3 + b^3$ is important because it allows you to simplify and manipulate polynomial expressions, which is a key skill in college algebra.
4. The sum of cubes formula can be extended to expressions with more than two variables, such as $a^3 + b^3 + c^3$, which can be factored as $(a + b + c)(a^2 - ab + b^2 - ac + bc - c^2)$.
5. Mastering the factorization of $a^3 + b^3$ and other polynomial expressions is essential for solving a variety of algebraic problems, including equations, inequalities, and word problems.

Review Questions

• Explain the significance of the expression $a^3 + b^3$ in the context of factoring polynomials.
  • The expression $a^3 + b^3$ is significant in the context of factoring polynomials because it can be factored using a special factorization method known as the sum of cubes formula. This formula states that $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. Mastering the factorization of $a^3 + b^3$ and other polynomial expressions is a crucial skill in college algebra, as it allows you to simplify and manipulate polynomial expressions, which is essential for solving a variety of algebraic problems.
• Describe the relationship between the sum of cubes formula and the factorization of polynomials with a difference of two cubes.
  • The sum of cubes formula can be used to factor polynomials with a difference of two cubes, such as $x^3 - y^3$. In this case, the polynomial can be factored as $(x - y)(x^2 + xy + y^2)$. This relationship is important because it demonstrates how the sum of cubes formula can be applied to a wider range of polynomial expressions, not just those with a sum of two cubes. Understanding this connection is essential for effectively factoring a variety of polynomial expressions in college algebra.
• Analyze how the sum of cubes formula can be extended to expressions with more than two variables, and explain the significance of this extension.
  • The sum of cubes formula can be extended to expressions with more than two variables, such as $a^3 + b^3 + c^3$, which can be factored as $(a + b + c)(a^2 - ab + b^2 - ac + bc - c^2)$. This extension is significant because it demonstrates the versatility and broader applicability of the sum of cubes formula. By understanding how to factor expressions with more than two variables using this formula, students can tackle a wider range of polynomial factorization problems in college algebra, which is crucial for developing a deep understanding of algebraic concepts and problem-solving skills.

Related terms

Polynomial:

A polynomial is an expression consisting of variables and coefficients, where the variables are raised to non-negative integer powers and combined using the operations of addition, subtraction, and multiplication.

Factoring:

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

Sum of Cubes Formula: The sum of cubes formula is a special factorization method used to factor the expression $a^3 + b^3$, where $a$ and $b$ are variables.