key term - Difference of Cubes

Definition

The difference of cubes is a special product in algebra where the difference between two cubes (the third power of a number) can be factored into a simpler expression. This concept is important in understanding how to multiply and factor polynomials, as well as solve polynomial equations.

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. The difference of cubes formula can be used to factor polynomials in the form $x^3 - a^3$.
3. Factoring the difference of cubes is a special case of the general strategy for factoring polynomials.
4. Difference of cubes factorization is often used to solve polynomial equations by breaking down the expression into simpler factors.
5. Recognizing the difference of cubes pattern is an important skill in multiplying and factoring polynomials.

Review Questions

• Explain how the difference of cubes formula can be used to factor a polynomial expression in the form $x^3 - a^3$.
  • The difference of cubes formula states that $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. To factor a polynomial expression in the form $x^3 - a^3$, we can apply this formula by letting $a = x$ and $b = a$. This gives us the factorization $x^3 - a^3 = (x - a)(x^2 + xa + a^2)$. This allows us to break down the original polynomial expression into a product of simpler factors.
• Describe how the difference of cubes concept is related to the general strategy for factoring polynomials.
  • The difference of cubes is a special case of the general strategy for factoring polynomials. The general strategy involves identifying common factors, recognizing special product patterns (such as the difference of squares or the difference of cubes), and using these patterns to factor the polynomial expression. The difference of cubes is one of the special product patterns that can be used to factor certain polynomial expressions more efficiently. Understanding the difference of cubes and how to apply the corresponding formula is an important part of the general factoring strategy.
• Explain how the difference of cubes factorization can be used to solve polynomial equations.
  • Polynomial equations in the form $x^3 - a^3 = 0$ can be solved by factoring the left-hand side using the difference of cubes formula. This gives us the equation $(x - a)(x^2 + xa + a^2) = 0$. To solve this equation, we can set each factor equal to zero and solve for $x$. The first factor $(x - a)$ gives us the solution $x = a$, while the second factor $(x^2 + xa + a^2)$ can be further factored using the quadratic formula to find the other two solutions. By breaking down the original polynomial equation using the difference of cubes factorization, we can more easily find the roots or solutions to the equation.

Related terms

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

Polynomial:

An expression consisting of variables and coefficients, where the variables are only raised to non-negative integer powers.

Factoring:

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