Problem 92 - College Algebra with Modeling and Visualization

Question

Factor the expression. \(x^{3}+1\)

Step-by-Step Solution

Answer

The expression \(x^3 + 1\) factors to \((x + 1)(x^2 - x + 1)\).

Step 1: Understanding the Expression

The expression we need to factor is \(x^3 + 1\). This is a sum of cubes.

Step 2: Recall the Formula for Sum of Cubes

The sum of cubes formula is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). In this case, \(a = x\) and \(b = 1\).

Step 3: Apply the Formula

Substitute \(x\) for \(a\) and \(1\) for \(b\) in the sum of cubes formula: \(x^3 + 1^3 = (x + 1)(x^2 - x \cdot 1 + 1^2)\).

Step 4: Simplify the Factors

Simplify the second part of the expression: \(x^2 - x \cdot 1 + 1^2 = x^2 - x + 1\).

Step 5: Write the Final Factored Form

The factored form of \(x^3 + 1\) is \((x + 1)(x^2 - x + 1)\).

Key Concepts

Sum of Cubes

Factoring expressions like \(x^3 + 1\) involves understanding the concept of the sum of cubes. The expression \(a^3 + b^3\) is a classic example of a sum of cubes. To factor it, we use the sum of cubes formula:

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

Here, \(a\) and \(b\) represent two numbers or algebraic expressions. Each term in the formula plays a specific role in breaking down the sum into factors. Notice that the first factor is a simple sum \((a + b)\), while the second factor is a bit more complex \((a^2 - ab + b^2)\).
When applying the formula to \(x^3 + 1\), we set \(a = x\) and \(b = 1\). This means the expression can be rewritten using the sum of cubes formula, leading to the factors \((x + 1)\) and \((x^2 - x + 1)\). Understanding the formula and how to apply it is key to efficiently factor sums of cubes.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. The expression \(x^3 + 1\) is a simple example, combining a variable \(x\) raised to the power of 3 with the constant 1. This kind of expression can be factored or simplified using a variety of algebraic techniques.
It's important to identify the form of an expression first. For instance, recognizing \(x^3 + 1\) as a sum of cubes allows us to utilize a specific factorization formula.

• Variables like \(x\) are placeholders that can represent numbers.
• Constants are fixed values, like the number 1 in our expression.
• Operations such as addition are the actions that combine these elements.

Understanding each component helps in applying the right factoring method, which in this case is the sum of cubes. By dissecting and comprehending the structure of algebraic expressions, the factorization process becomes more intuitive.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial, like \(x^3 + 1\), into simpler, non-divisible parts known as factors. Factoring is essential because it simplifies polynomials, making them easier to manage in algebraic operations and solving equations.

• Factors are expressions that, when multiplied together, give the original polynomial.
• This process reduces the complexity of the polynomial.

In the exercise, \(x^3 + 1\) is factored into \((x + 1)(x^2 - x + 1)\). Each of these factors can be handled separately, allowing us to explore solutions or further simplifications.
To factor polynomials, it's crucial to recognize patterns and apply appropriate formulas like the sum of cubes. This way, the polynomial is decomposed into fundamental parts so it can be analyzed or solved more effectively.
Whether dealing with simple polynomials or more complex ones, mastering factorization techniques enhances problem-solving skills and deepens understanding of algebraic structures.