Problem 128 - College Algebra Essentials

Question

Explain how to factor $$x^{3}+1$$

Step-by-Step Solution

Answer

The factored form of the expression \(x^{3} + 1\) is \((x + 1)(x^{2} - x + 1)\)

Step 1: Identify the form of the expression

Recognize the expression \(x^{3} + 1\) as a sum of cubes, in this case, \(x^{3} + 1^{3}\). This corresponds to the formula for the sum of cubes, which is \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\)

Step 2: Substitute the values of a and b

In our case, \(a = x\) and \(b = 1\). Substituting these values into the formula gives \((x + 1)(x^{2} - x*1 + 1^{2})\) or \((x + 1)(x^{2} - x + 1)\)

Step 3: Simplify the expression

The expression simplifies to \((x + 1)(x^{2} - x + 1)\),if needed, this can also be written as \(x^{3}+1 = (x + 1)(x^{2} - x + 1)\) to show the initial form and the factored form of the expression together.

Key Concepts

Sum of Cubes

In algebra, a common task is to factor expressions, and one special type of expression is the "sum of cubes." The sum of cubes refers to an expression where two cubed numbers are added together, like in the formula \(a^3 + b^3\). Recognizing this form is key to correctly breaking down and simplifying the expression.

For example, if you come across the expression \(x^3 + 1\), it can be identified as a sum of cubes because it adds \(x^3\) (a cube of \(x\)) and \(1^3\) (a cube of \(1\)).

The formula to factor any sum of cubes \(a^3 + b^3\) is given by:
\[(a + b)(a^2 - ab + b^2)\]
This formula helps transform a complex cubic expression into simpler factors, making it easier to solve, analyze, and understand.

Factored Form

To reach the factored form of a sum of cubes, such as \(x^3 + 1\), we apply the sum of cubes formula. The objective is to express the cubic expression as a product of two simpler components. This is beneficial because factored expressions make it easier to find roots and solve equations.

Using the earlier mentioned formula:

• Identify \(a\) and \(b\). Here, \(a = x\) and \(b = 1\).
• Substitute these values into the sum of cubes formula: \((x + 1)(x^2 - x \cdot 1 + 1^2)\).
• Simplify to obtain \((x + 1)(x^2 - x + 1)\).

This provides a product of linear and quadratic terms, making the expression manageable and simplifying further operations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They can represent real-world problems or purely mathematical concepts and are fundamental to understanding the language of algebra.

Expressions similar to \(x^3 + 1\) are categorized as polynomial expressions since they consist of terms with non-negative integer exponents. Polynomials like these are building blocks for more complex algebraic tasks.

In any polynomial expression:

• The degree is determined by the highest power of the variable—in \(x^3 + 1\), the degree is 3.
• Terms are separated by addition or subtraction signs, and each term can include coefficients and variables.
• Simplifying and factoring these expressions is often necessary for solving equations, understanding functions, and contributing to broader mathematical discussions.

By mastering algebraic expressions, students gain a powerful toolkit for tackling a wide variety of mathematical challenges.