Sum of Cubes

Definition

The sum of cubes is a mathematical expression that represents the sum of the cubes of two or more numbers. It is an important concept in various areas of mathematics, including factoring polynomials, 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 sum of cubes can be expressed as $a^3 + b^3$, where $a$ and $b$ are two numbers.
2. The sum of cubes can be factored using the formula $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.
3. The sum of cubes is a special case of the sum of powers, where the exponent is 3.
4. The sum of cubes is often used in solving certain types of equations, such as those involving perfect cubes.
5. The sum of cubes is a key concept in understanding the behavior of power functions and polynomial functions, particularly in the context of graphing and analyzing their properties.

Review Questions

• Explain how the sum of cubes can be used in the factorization of polynomials.
  • The sum of cubes formula, $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$, can be used to factor certain polynomial expressions that are the sum of cubes. For example, the polynomial $x^3 + 8$ can be factored as $(x + 2)(x^2 - 2x + 4)$ by recognizing that $x^3 + 8$ is the sum of cubes, with $a = x$ and $b = 2$. This factorization technique can be applied to a variety of polynomial expressions that can be expressed as the sum of cubes.
• Describe how the sum of cubes is used in solving certain types of equations.
  • The sum of cubes can be used to solve equations that involve perfect cubes. For instance, an equation like $x^3 + 27 = 0$ can be solved by recognizing that $x^3 + 27$ is the sum of cubes, with $a = x$ and $b = 3$. Using the sum of cubes formula, we can factor the left-hand side of the equation to obtain $(x + 3)(x^2 - 3x + 9) = 0$, which leads to the solution $x = -3$. This method of solving equations by factoring the sum of cubes is particularly useful in certain types of polynomial and radical equations.
• Analyze how the sum of cubes relates to the properties of power functions and polynomial functions.
  • The sum of cubes is closely related to the behavior of power functions and polynomial functions. For power functions, the sum of cubes formula can be used to understand the relationship between the cubes of different inputs, which is relevant in analyzing the properties of these functions, such as their end behavior and the locations of their local extrema. Similarly, in the context of polynomial functions, the sum of cubes formula can provide insights into the factorization and graphing of polynomials that include the sum of cubes as a term. Understanding the sum of cubes is therefore crucial in developing a deeper understanding of the properties and characteristics of power functions and polynomial functions.

Related terms

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

Difference of Cubes: The difference of cubes is a special case of the sum of cubes, where the expression represents the difference between the cubes of two numbers, such as $a^3 - b^3$.

Factoring Polynomials: Factoring polynomials is the process of expressing a polynomial as a product of simpler polynomials, which can involve identifying and using the sum or difference of cubes.