Problem 52 - College Algebra

Question

In Exercises \(49-56,\) factor using the formula for the sum or difference of two cubes. $$x^{3}-27$$

Step-by-Step Solution

Answer

The factored form of \(x^{3}-27\) is \((x-3)(x^{2}+3x+9)\).

Step 1: Identify the Cubes

In the expression \(x^{3}-27\), \(x^{3}\) is the first cube and \(27\) is the second cube. So, \(a^{3}=x^{3}\) which means \(a=x\), and \(b^{3}=27\) which means \(b=3\) as \(3^{3}\) equals to \(27\).

Step 2: Substitute \(a\) and \(b\) into the Formula

Substitute \(a=x\) and \(b=3\) into the formula for the difference of two cubes: \((a-b)(a^{2}+ab+b^{2})\). So, the factored form is \((x-3)(x^{2}+3x+9)\).

Step 3: Check the Factored Expression

One can check the factored expression by expanding \((x-3)(x^{2}+3x+9)\) and verifying that it equals to the original expression \(x^{3}-27\). Expanding \((x-3)(x^{2}+3x+9)\) yields \(x^{3}-27\), which confirms that the factored form is correct.

Key Concepts

Understanding Cubes

Cubes are a fascinating concept in mathematics, representing the multiplication of a number by itself twice. In simple terms, when we say a number is cubed, it means it is raised to the power of three.

• For example, when we say \( x^3 \), it implies \( x \times x \times x \).
• Similarly, \( 3^3 \) means \( 3 \times 3 \times 3 \), which equals \( 27 \).

Cubes can be visualized as a cube in three-dimensional geometry, where the length, width, and height are all equal. Understanding cubes is essential, especially when factoring polynomials, as they lay the foundation for solving more complex equations. It is important to know the basic cubes like \( 2^3 = 8 \), \( 4^3 = 64 \), and so on, which makes identifying cubes in expressions much simpler.Comprehending cubes makes it easier to detect them within algebraic expressions, setting the stage for operations like factorization.

Difference of Cubes Explained

The difference of cubes refers to a specific type of expression where one cubed term is subtracted from another. When we come across expressions like \( x^3 - 27 \), we can treat them as the difference of two cubes. This is a special algebraic form with a distinct factorization method. Let's break down the formula for the difference of cubes:Given two cubes, say \( a^3 \) and \( b^3 \), the difference is expressed in a formula as:\[(a^3 - b^3) = (a - b)(a^2 + ab + b^2)\]

Steps to Factor a Difference of Cubes:

• Identify the terms that are perfect cubes in your expression.
• For \( x^3 - 27 \), we recognize that \( x \) and \( 3 \) are our bases because \( 27 \) is \( 3^3 \).
• Inserting these into the formula, we get \((x - 3)(x^2 + 3x + 9)\).

This formula effectively breaks the difference of cubes into a product of a linear polynomial and a quadratic polynomial, simplifying the equation for further analysis or solving.

The Art of Factorization

Factorization is the process of breaking down an algebraic expression into the simplest polynomials that can multiply to give the original expression. It’s a bit like dismantling something complicated into all of its individual pieces so you can see how it was constructed.

Why Factorization is Important:

• It simplifies complex algebraic expressions, making them easier to work with.
• Factorization helps in solving algebraic equations by finding the roots of the polynomial.
• It provides insights into the properties of mathematical relationships and aids in the understanding of polynomial functions.

When dealing with polynomials like \( x^3 - 27 \), recognizing and applying the difference of cubes rule is a crucial factorization technique. By transforming \( x^3 - 27 \) into \( (x - 3)(x^2 + 3x + 9) \), we make solving for \( x \) or analyzing the polynomial function far more straightforward.Mastering factorization strategies like these enhances problem-solving skills and boosts overall mathematical comprehension, equipping learners to tackle more challenging equations in the future.