Problem 55 - Elementary and Intermediate Algebra

Question

Factor each sum of cubes. \(8 r^{3}+s^{3}\)

Step-by-Step Solution

Answer

\((2r + s)(4r^2 - 2rs + s^2)\) is the factored form.

Step 1: Recognize the Sum of Cubes Formula

The sum of cubes formula is given by: \[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \] Identify the expressions as cubes. Here, \(8r^3\) can be written as \((2r)^3\) and \(s^3\) is already a cube of \(s\).

Step 2: Identify 'a' and 'b'

From the expression \(8r^3 + s^3\), compare with \(a^3 + b^3\). So, \(a = 2r\) and \(b = s\).

Step 3: Apply the Sum of Cubes Formula

Substitute \(a\) and \(b\) into the formula:\[ (2r + s)((2r)^2 - (2r)(s) + (s)^2) \]

Step 4: Simplify the Expression

Calculate the individual terms:- \((2r)^2 = 4r^2\)- \((2r)(s) = 2rs\)- \((s)^2 = s^2\)So the expression becomes:\[ (2r + s)(4r^2 - 2rs + s^2) \]

Key Concepts

Sum of Cubes Formula

Understanding the sum of cubes formula is essential when factoring specific types of polynomial expressions. The sum of cubes formula is \[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \]This formula allows us to break down and simplify expressions that are the sum of two cubes. To use this formula effectively:

• Identify each part of the expression as a cube. For example, for the expression \(8r^3 + s^3\), note that \(8r^3\) is the cube of \(2r\) and \(s^3\) is the cube of \(s\).
• Define \(a\) and \(b\) based on these cubes, where \(a = 2r\) and \(b = s\).

Substitute these values into the formula and simplify, as shown in the problem solution. Recognizing the structure of cube expressions is the key to utilizing this formula effectively.

Polynomial Expressions

Polynomial expressions are mathematical expressions made up of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents. Factoring polynomials, such as the sum of cubes, is a way to simplify them. It involves representing a polynomial as a product of its factors, which can be numbers, variables, or other polynomials.
In the context of the provided expression \(8r^3 + s^3\):

• Identify terms separately: \(8r^3\) and \(s^3\).
• Recognize that polynomial expressions can be simplified using identities and formulas, like the sum of cubes.
• Understand that the goal of factoring is to express the polynomial in a simpler, factored form, \((2r + s)(4r^2 - 2rs + s^2)\), which is easier to work with in subsequent calculations.

Breaking down and simplifying polynomial expressions helps make complex calculations more manageable.

Algebraic Identities

Algebraic identities are equations that are true for all values of the variables involved. They are powerful tools in algebra that allow us to simplify and manipulate expressions. The sum of cubes formula \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\) is one such identity.
To effectively use algebraic identities:

• Understand the structure of the identity and how it relates to the expression you are working with.
• Apply the correct identity based on the form of the polynomial. For example, when dealing with a sum of two cubes, the sum of cubes identity applies.
• Simplify the expression by substituting the identified components, such as \(a = 2r\) and \(b = s\), into the identity. Then carry out the multiplication and simplification.

Using algebraic identities like the sum of cubes helps to transform complex polynomial expressions into simpler, more workable forms, providing clarity and ease in computations.