key term - Sum and Difference of Cubes

Definition

The sum and difference of cubes refers to two special algebraic identities that allow for the factorization of expressions involving the sum or difference of two cubes. These identities provide a useful tool for simplifying and factoring polynomial expressions.

5 Must Know Facts For Your Next Test

1. The sum of cubes identity states that $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.
2. The difference of cubes identity states that $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.
3. These identities can be used to factor expressions involving the sum or difference of two cubes.
4. Factoring expressions using the sum and difference of cubes identities can simplify algebraic expressions and make them easier to work with.
5. The sum and difference of cubes identities are particularly useful in the context of 7.4 Factor Special Products, as they provide a systematic way to factor certain types of polynomial expressions.

Review Questions

• Explain how the sum of cubes identity can be used to factor an expression of the form $a^3 + b^3$.
  • The sum of cubes identity states that $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. To factor an expression of the form $a^3 + b^3$, we can identify the values of $a$ and $b$ in the expression and then apply the identity. For example, if the expression is $x^3 + 8$, we can see that $a = x$ and $b = 2$, so the factorization would be $(x + 2)(x^2 - 2x + 4)$.
• Describe the relationship between the sum and difference of cubes identities and the concept of factoring special products.
  • The sum and difference of cubes identities are closely related to the topic of factoring special products, as they provide a systematic way to factor certain types of polynomial expressions. These identities allow us to recognize and factor expressions involving the sum or difference of two cubes, which are considered special products. By understanding and applying these identities, we can efficiently factor polynomial expressions that would otherwise be more challenging to factor using general factorization methods.
• Analyze how the structure of the sum and difference of cubes identities can be used to simplify and manipulate algebraic expressions.
  • The structure of the sum and difference of cubes identities, $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ and $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, reveals important information about the relationships between the terms. By recognizing the presence of a sum or difference of cubes in an expression, we can apply the appropriate identity to factor the expression and simplify it. This can involve identifying the values of $a$ and $b$, as well as manipulating the expression to match the form of the identity. Understanding the structure of these identities and how to apply them is crucial for effectively factoring and simplifying polynomial expressions.

Related terms

Cube: The result of multiplying a number by itself three times, represented as $x^3$.

Factorization: The process of expressing a polynomial as a product of simpler polynomial factors.

Algebraic Identities:

Equations that are true for all values of the variables involved, used to simplify algebraic expressions.