Problem 75 - College Algebra

Question

Factor completely, or state that the polynomial is prime. $$ x^{3}-4 x $$

Step-by-Step Solution

Answer

The completely factored form of the given polynomial \(x^{3}-4x\) is \(x(x+2)(x-2)\).

Step 1: Identify Common Factors

To begin with, we examine the polynomial \(x^{3}-4x\) for any common factors across all terms. Here, 'x' is common to both terms, so we can begin by factoring it out of the expression.

Step 2: Factor out the Common Factors

After identifying the common factor, we will factor it out of the expression. Factoring 'x' out of \(x^{3}-4x\) we get: \(x(x^{2}-4)\)

Step 3: Further Factorize the Expression if Possible

After factoring out 'x', the remaining expression is \(x^{2}-4\). This is a difference of squares, which further factorizes to \((x+2)(x-2)\).

Step 4: Write the Completely Factored Form of the Polynomial

Combining steps 2 and 3, the completely factored form of the given polynomial \(x^{3}-4x\) is: \(x(x+2)(x-2)\). This is the simplest form of the given polynomial.

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is crucial in comprehending the basics of algebra. An algebraic expression is a mathematical phrase that can consist of numbers, variables (like x or y), and arithmetic operations (such as addition, subtraction, multiplication, and division). For example, in the given exercise, the expression is represented by \(x^{3}-4x\), which includes a variable 'x' raised to a power and a multiplication of 'x' by a number.

To work effectively with algebraic expressions, one must be familiar with the rules and properties of arithmetic operations, such as the distributive property, which can be applied when factoring out common factors from an expression. In the given example, 'x' is the common factor identified in both terms of the polynomial \(x^{3}-4x\), allowing for the simplification and easier understanding of the relationship between the terms.

Common Factors

When we talk about common factors in the context of algebra, we refer to the shared factors found in each term of a polynomial. Factoring in algebra is a critical skill that involves rewriting expressions as the product of their factors. It makes complex problems simpler and more manageable.

In the exercise \(x^{3}-4x\), we see that 'x' appears in both terms of the polynomial and can be factored out. By doing so, we create a simpler expression. Factoring out 'x' results in \(x(x^{2}-4)\), revealing the underlying structure of the polynomial and setting the stage for even further simplification by recognizing and exploiting more complex patterns, such as the difference of squares present in this example.

Difference of Squares

The difference of squares is a pattern in algebra where a two-term polynomial is the difference (subtraction) of two perfect squares. This pattern leads to a special factoring formula: \(a^{2} - b^{2} = (a + b)(a - b)\). Recognizing this pattern is a powerful tool for simplification.

After factoring out the common 'x' from the original exercise's polynomial, we are left with the difference of squares, \(x^{2}-4\). According to the formula, it can be broken down into \((x+2)(x-2)\). This final step reveals the fully factored form of the original expression, making it evident how underlying structures like the difference of squares can significantly simplify polynomial expressions. This knowledge is a cornerstone in algebra that helps students solve a variety of problems efficiently.