key term - Difference of Cubes

Definition

The difference of cubes is a special case of polynomial factorization where a polynomial expression can be factored by recognizing the difference between two cubes. This factorization technique is useful in solving certain types of equations and understanding the behavior of power functions and polynomial functions.

5 Must Know Facts For Your Next Test

1. The difference of cubes is represented by the expression $a^3 - b^3$, where $a$ and $b$ are any real numbers.
2. The difference of cubes can be factored using the formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.
3. Recognizing the difference of cubes pattern is useful in solving certain types of equations, such as those involving perfect cubes.
4. The difference of cubes factorization can be applied to power functions and polynomial functions to understand their behavior and properties.
5. The difference of cubes factorization is a specific case of the general binomial factorization formula: $a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + ... + ab^{n-2} + b^{n-1})$.

Review Questions

• Explain how the difference of cubes factorization can be used to solve certain types of equations.
  • The difference of cubes factorization can be used to solve equations involving perfect cubes, such as $x^3 - a^3 = 0$. By recognizing the difference of cubes pattern, we can factor the left-hand side of the equation and then solve for the variable $x$ by setting each factor equal to zero. This approach simplifies the equation and makes it easier to find the solutions.
• Describe how the difference of cubes factorization can be applied to the study of power functions and polynomial functions.
  • The difference of cubes factorization can provide insights into the behavior and properties of power functions and polynomial functions. For example, in the power function $f(x) = x^3 - a^3$, the difference of cubes factorization can be used to understand the factors that influence the function's graph, such as its zeros, local extrema, and asymptotic behavior. Similarly, in the study of polynomial functions, the difference of cubes factorization can be a useful tool in determining the roots, factors, and other characteristics of the polynomial.
• Analyze the general binomial factorization formula and explain how it relates to the specific case of the difference of cubes.
  • The general binomial factorization formula, $a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + ... + ab^{n-2} + b^{n-1})$, encompasses the difference of cubes factorization as a specific case. When $n = 3$, the general formula simplifies to the difference of cubes factorization: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. This demonstrates how the difference of cubes factorization is a specialized application of the broader binomial factorization principle, which can be applied to a wide range of polynomial expressions.

Related terms

Cube: A cube is a three-dimensional geometric shape where all sides are equal in length, resulting in a perfect cube.

Factorization:

Factorization is the process of breaking down a polynomial expression into a product of simpler polynomial expressions.

Power Function:

A power function is a function of the form $f(x) = x^n$, where $n$ is a real number, and it exhibits unique characteristics based on the value of $n$.