Problem 61 - College Algebra and Calculus: An Applied Approach

Question

Completely factor the expression.\(x^{4}-4 x^{3}+x^{2}-4 x\)

Step-by-Step Solution

Answer

The factored form of the expression \(x^{4}-4 x^{3}+x^{2}-4 x\) is \((x^{3}+x)(x - 4)\)

Step 1: Identify the terms

The expression can be rewritten as \(x^{4}+(-4x^{3})+x^{2}+(-4x)\), where \(x^{4}\), \(-4x^{3}\), \(x^{2}\), and \(-4x\) are the terms.

Step 2: Group the terms

Separate this into two groups of terms. Group the terms as \((x^{4} - 4x^{3}) + (x^{2} - 4x)\)

Step 3: Find common factors from each group

Factor out the greatest common factor from each group: \(x^{3}(x - 4) + x(x - 4)\). Now there are two terms, \(x^{3}(x - 4)\) and \(x(x - 4)\). Note that \(x - 4\) is a common factor.

Step 4: Factor out common factor

Now we will factor out the common binomial factor from the expression \(x^{3}(x - 4) + x(x - 4)\) to get final factored form of the polynomial: \((x^{3}+x)(x - 4)\). This is the complete factored form of the polynomial.

Key Concepts

Polynomial Expressions

A polynomial expression is a combination of variables and constants using addition, subtraction, multiplication, and non-negative integer exponents. In our exercise, the polynomial is given by \(x^{4} - 4x^{3} + x^{2} - 4x\). It consists of four terms, each involving a power of \(x\). This polynomial is in its expanded form, making it crucial to simplify or factor it for deeper understanding or solution finding. The basic operations involved in dealing with polynomial expressions include:

• Addition and subtraction, which combine or separate terms.
• Multiplication, which distributes one polynomial over another.
• Finding the greatest common factor (GCF) to simplify expressions.

Remember, when you look at polynomial expressions, check each term’s degree to identify the highest power. The degree helps determine its properties and is a starting point for many algebraic processes.

Common Factor Extraction

Common factor extraction is a crucial step in simplifying polynomial expressions. It's all about identifying a common term in each part of your polynomial and pulling it out in front. In the exercise, the expression is split into two groups: \((x^{4} - 4x^{3})\) and \((x^{2} - 4x)\). The key to simplifying these groups is spotting the greatest common factor (GCF). For \(x^{4} - 4x^{3}\), \(x^{3}\) is the GCF because it appears in both terms. Similarly, for \(x^{2} - 4x\), \(x\) is the GCF. Once you identify these, factor them out of each group, which changes the expression to \(x^{3}(x - 4) + x(x - 4)\). This step is essential because it sets the foundation for further factorization, revealing further simplifications or necessary steps needed to reach the complete factorized form.

Grouping Method

The grouping method is designed to help identify and extract commonalities from a polynomial, especially when direct factorization is not immediately obvious. In our exercise, the polynomial is grouped into two parts: \((x^{4} - 4x^{3})\) and \((x^{2} - 4x)\). This grouping helps organize the polynomial into manageable pieces. Once the polynomial is split into groups, each section is simplified by extracting the common factors. In this case:

• \(x^{3}\) is factored from the first group \((x^{4} - 4x^{3})\).
• \(x\) is factored from the second group \((x^{2} - 4x)\).

The result is \(x^{3}(x - 4) + x(x - 4)\). By restructuring the expression, the grouping method often makes the final factorization more clear. Here, it instantly revealed \(x - 4\) as a common factor between the grouped parts.

Algebraic Manipulation

Algebraic manipulation involves altering the form of an algebraic expression to solve it more easily or make it clearer. The aim is often to simplify an expression or solve an equation. In this exercise, algebraic manipulation finalizes the factorization process. After recognizing the common factor \(x - 4\), use algebraic manipulation to factor it out of the entire expression \(x^{3}(x - 4) + x(x - 4)\), resulting in \((x^{3} + x)(x - 4)\). Algebraic manipulation requires a good understanding of algebra rules and properties like distribution, factoring, and grouping. It's about rearranging terms, and using brackets effectively to reach a form that is simpler or solves the problem. This process enhances your ability to see relationships between terms, and further develops problem-solving skills. With practice, algebraic manipulation becomes a powerful tool in tackling a wide array of mathematical problems.