Difference of Cubes

Definition

The difference of cubes is a special type of polynomial expression that can be factored using a specific formula. It refers to the difference between two perfect cubes, which are the results of raising a number to the third power.

5 Must Know Facts For Your Next Test

1. The difference of cubes formula is: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.
2. Factoring the difference of cubes is a specific application of the general strategy for factoring polynomials.
3. Identifying a difference of cubes expression is the first step in applying the difference of cubes formula to factor the polynomial.
4. The difference of cubes formula can be used to factor polynomials with higher-degree terms, such as $x^6 - y^6$.
5. Mastering the difference of cubes factorization technique is important for solving more complex polynomial equations and expressions.

Review Questions

• Explain the difference of cubes formula and how it is used to factor polynomials.
  • 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 polynomial expressions that represent the difference between two perfect cubes. By identifying the a and b terms in the expression, the polynomial can be factored into the product of two binomial factors using this formula. This factorization technique is a specific application of the general strategy for factoring polynomials.
• Describe how the difference of cubes formula can be used to factor more complex polynomial expressions.
  • The difference of cubes formula can be extended to factor polynomials with higher-degree terms, such as $x^6 - y^6$. In this case, the polynomial can be rewritten as $(x^3)^2 - (y^3)^2$, which is in the form of a difference of cubes. Applying the difference of cubes formula, the expression can be factored as $(x^3 - y^3)(x^3 + y^3)$. This demonstrates the versatility of the difference of cubes factorization technique in handling more complex polynomial expressions.
• Analyze the importance of mastering the difference of cubes factorization technique within the context of the general strategy for factoring polynomials.
  • Mastering the difference of cubes factorization technique is crucial within the broader context of the general strategy for factoring polynomials. The ability to recognize and apply the difference of cubes formula is a key skill that allows students to efficiently factor a wide range of polynomial expressions. This technique not only simplifies the factorization process but also lays the foundation for understanding more advanced polynomial manipulation and problem-solving. Proficiency in difference of cubes factorization demonstrates a deeper understanding of polynomial structure and the strategic approaches necessary for successful polynomial factorization.

Related terms

Perfect Cube: A number that is the result of raising another number to the third power, such as $8 = 2^3$ or $27 = 3^3$.

Factoring: The process of breaking down a polynomial expression into a product of simpler polynomial expressions.

Polynomial Expression: An expression that contains one or more variables and is made up of terms that are products of variables raised to non-negative integer powers and coefficients.