Problem 79 - Introductory Algebra for College Students

Question

Factor using the formula for the sum or difference of two cubes. $$x^{3}+1$$

Step-by-Step Solution

Answer

The factorization of \(x^{3} + 1\) is \((x + 1)(x^{2} - x + 1)\).

Step 1: Identify the a and b values

In the expression \(x^{3} + 1\) , we have \(a = x\) and \(b = 1\).

Step 2: Apply the formula for the sum of cubes

Using the sum of cubes formula \((a + b)(a^{2} - ab + b^{2})\), we can substitute \(a = x\) and \(b = 1\).

Step 3: Simplify the expression

By substituting into the formula we get \((x + 1)(x^{2} - x + 1)\).

Key Concepts

Sum of Cubes Formula

Understanding the sum of cubes formula is critical when you are dealing with factoring algebraic expressions that appear as the sum of two perfect cubes. The sum of cubes formula is an important tool and it's given by \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). It allows us to factor an expression that at first glance does not seem factorable.

For instance, in the expression \(x^3 + 1\), it may not be obvious that this is a sum of cubes. However, by recognizing that \(1\) can be written as \(1^3\), we see that we indeed have two cubes: \(x^3\) and \(1^3\). Using the formula, we pair \(x\) with \(a\) and \(1\) with \(b\) to rewrite the expression as a product of two binomials. This results in the factored form \( (x + 1)(x^2 - x + 1) \).

It's a clever trick that often comes in handy, especially when solving polynomial equations or simplifying complex algebraic fractions.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra. They consist of numbers, variables (like \(x\) or \(y\)), and arithmetic operations (+, -, *, /). The expression \(x^3 + 1\) is an example of an algebraic expression, where \(x\) is the variable, and 1 is the constant.

An important skill in algebra is identifying the form of an algebraic expression, which can often provide clues to which factoring technique should be applied. In our example, recognizing the expression as a sum of cubes leads directly to the appropriate factoring strategy.

Understanding how to manipulate these expressions, whether it's through expanding, factoring, or simplifying, is crucial for solving equations, understanding function behavior, and analyzing patterns or relationships between variables.

Polynomial Factoring

Polynomial factoring is a process used in algebra to break down a polynomial into simpler 'factorable' components, which when multiplied together give back the original polynomial. It's analogous to factoring numbers, like expressing 12 as \(2 \times 2 \times 3\), but with algebraic expressions.

Considering our previous example, \(x^3 + 1\), we use the sum of cubes formula for factoring. However, other common factoring techniques include finding common factors, using the difference of squares formula, or factoring by grouping.

Often, factoring is used to find the roots, or 'zeros', of a polynomial function. In other words, it helps to find the values for the variable where the function's output is zero. Knowing various factoring techniques is essential for solving higher degree polynomials, which frequently occur in algebra and calculus.