Problem 23 - Algebra and Trigonometry

Question

\(21-28\) Use a Factoring Formula to factor the expression. $$ 27 x^{3}+y^{3} $$

Step-by-Step Solution

Answer

\((3x + y)(9x^2 - 3xy + y^2)\)

Step 1: Identify the Formula

The expression we are given, \(27x^3 + y^3\), involves two cubes. This fits the form of a sum of cubes, which can be factored using the formula: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Identify \(a\) and \(b\) such that \(a^3 = 27x^3\) and \(b^3 = y^3\).

Step 2: Find Terms a and b

From the expression \(a^3 = 27x^3\), we find \(a = 3x\) since \((3x)^3 = 27x^3\). Similarly, from \(b^3 = y^3\), we find \(b = y\).

Step 3: Substitute and Apply the Formula

Substitute \(a = 3x\) and \(b = y\) into the sum of cubes formula: \((a + b)(a^2 - ab + b^2)\). This gives us \((3x + y)((3x)^2 - (3x)(y) + y^2)\).

Step 4: Simplify the Expression

Calculate each part of the second factor: - \((3x)^2 = 9x^2\)- \(3x \cdot y = 3xy\)- \(y^2\) remains as it is.Substitute these into the expression: \((3x + y)(9x^2 - 3xy + y^2)\).

Step 5: Write the Final Factored Form

The expression \(27x^3 + y^3\) is factored as \((3x + y)(9x^2 - 3xy + y^2)\).

Key Concepts

Sum of Cubes

The **sum of cubes** is a special type of algebraic expression and can be factored using a specific formula. It involves expressions of the form \( a^3 + b^3 \). This expression is the sum of two cube terms. Why is factoring it important? Because it simplifies the expression and can be useful in solving equations or simplifying algebraic expressions. The sum of cubes has a straightforward factoring formula:

• \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)

This formula splits the expression into a product of two binomials, which are easier to manage. In practice, you're looking for two terms, \( a \) and \( b \), whose cubes make up the original expression. Once you identify these terms, you substitute them into the formula and simplify.

Algebra

**Algebra** is a branch of mathematics that deals with symbols and rules for manipulating those symbols. In algebra, we're often working with variables, constants, coefficients, and algebraic expressions. These elements are essential in forming equations, solving problems, and understanding patterns in mathematics.
Algebra provides useful tools to handle one or more operations, making it possible to construct equations that represent real-world situations or mathematical problems. Factoring is a critical skill in algebra, used to simplify expressions and solve equations.
In this exercise, we're using algebraic skills to factor a polynomial expression, specifically a sum of cubes. Understanding the algebraic rules and operations allows us to apply the sum of cubes formula accurately and effectively.

Factoring Formulas

**Factoring formulas** are specific algebraic expressions that allow you to transform complex polynomials into simpler, more manageable pieces. They're like templates that help you rewrite an expression in a factored form.

• For example, the sum of cubes formula \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \) helps break down an expression into two binomials.
• These formulas provide shortcuts for recognizing patterns and structures in polynomials, which makes solving equations faster and more intuitive.

To use a factoring formula effectively, such as in the original problem \( 27x^3 + y^3 \), you need to recognize the pattern in the expression. Then, identify the terms \( a \) and \( b \) that fit the formula. After substituting these terms into the factoring formula, you'll be able to convert the complex expression into a product of simpler parts, making it easier to work with or solve.