Problem 26 - Elementary and Intermediate Algebra

Question

Factor. $$ 8 u^{3}+w^{3} $$

Step-by-Step Solution

Answer

The expression factors to \((2u+w)(4u^2 - 2uw + w^2)\).

Step 1: Recognize the Sum of Cubes

The expression \(8u^3 + w^3\) can be identified as a sum of cubes, since \(8u^3\) is \((2u)^3\) and \(w^3\) is \(w^3\). The general factorization for a sum of cubes, \(a^3 + b^3\), is \((a+b)(a^2-ab+b^2)\).

Step 2: Apply the Sum of Cubes Formula

Assign \(a = 2u\) and \(b = w\). Substitute these into the formula for the sum of cubes: \((a+b)(a^2-ab+b^2)\), resulting in \((2u+w)((2u)^2 - (2u)(w) + w^2)\).

Step 3: Simplify Inside the Parentheses

Calculate \((2u)^2\), which is \(4u^2\). Calculate \((2u)(w)\), which gives \(2uw\). Now, substitute these expressions back into the expression: \((2u+w)(4u^2 - 2uw + w^2)\).

Step 4: Write the Final Factored Form

Combine the results to express the original expression as the product of two binomials: \((2u+w)(4u^2 - 2uw + w^2)\).

Key Concepts

Sum of Cubes

When working with polynomials, one interesting pattern you might encounter is the "sum of cubes." This occurs when you have an expression of the form \(a^3 + b^3\). Recognizing this pattern allows you to simplify the expression using a special formula.

For the sum of cubes, the factorization formula is:

• \((a + b)(a^2 - ab + b^2)\).

This formula allows you to break down the sum of cubes expression into more manageable parts, making it easier to work with.

In this context, given the expression \(8u^3 + w^3\), you'll observe that \(8u^3\) can be rewritten as \((2u)^3\), meaning that \(a = 2u\) and \(b = w\). Applying the sum of cubes factorization saves time and effort in simplifying such expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition, subtraction, multiplication, and division). Understanding how to manipulate these expressions is crucial for solving algebra problems, including those involving polynomials.

Each part of an algebraic expression can have:

• Variables: these are symbols, like \(u\) and \(w\), that represent numbers.
• Coefficients: these are numbers that multiply the variables, such as 2 in \(2u\).
• Constants: these are standalone numbers without variables.

Expressions can be simplified, factored, or expanded depending on the problem. In the context of the original problem of \(8u^3 + w^3\), the task was to factor the polynomial using the sum of cubes approach, illustrating the power of recognizing algebraic forms.

Polynomial Factorization

Factorization is a key process in working with polynomials. It involves expressing a polynomial as a product of simpler polynomials, making it easier to solve equations or analyze their properties.

The goal is to rewrite a given polynomial into factors, which are polynomials that can be multiplied to get the original polynomial. For example, the polynomial \(8u^3 + w^3\) can be factored into \((2u + w)(4u^2 - 2uw + w^2)\).

This transformation utilizes strategic formulas, like the sum of cubes, to recognize structures and simplify expressions efficiently. Factorization not only simplifies the expression but also reveals useful properties and potential solutions, making it a vital skill in algebra.