Problem 7 - Precalculus : Building Concepts and Connections

Question

Complete them to review topics relevant to the remaining exercises. Factor: \(x^{3}-3 x^{2}-4 x\)

Step-by-Step Solution

Answer

The factored form of the expression \(x^{3}-3 x^{2}-4 x\) is \(x (x-4) (x+1)\).

Step 1: Identify Common Factors

The first step in factoring is to identify if there is a common factor between the terms. In this case, \(x\) is a common factor among all terms. We, therefore, factor out \(x\).

Step 2: Factor Out the Common Factor

Factoring out \(x\) from all terms gives us \(x (x^{2}-3x-4)\). This leaves us with a quadratic equation inside the brackets.

Step 3: Factor the Quadratic Equation Inside the Brackets

Factor the quadratic equation inside the brackets \(x^{2}-3x-4\). The solutions to this equation are the factors of -4 that sum up to -3. These are -4 and 1. Therefore, the factored form of the quadratic equation is \((x-4) (x+1)\).

Step 4: Combine the Common Factor with the Factors of the Quadratic Expression

Combine the common factor from step 2 and the factored form of the quadratic equation from step 3 to get the final factored form of the given expression. This yields \(x (x-4) (x+1)\)

Key Concepts

Common Factor Identification

Understanding how to identify common factors within algebraic expressions is a fundamental skill in algebra that simplifies complex problems. When you look at a polynomial like \(x^{3}-3x^{2}-4x\), the first step is to observe the terms and find any commonalities. Each term in this polynomial includes the variable \(x\), which is known as the common factor.

To factor out this common factor, you'll divide each term by \(x\), effectively simplifying the expression and setting the stage for further factoring. This step not only makes the equation more manageable but also gives us a clearer picture of what we're working with by removing the shared component from each part of the equation. By identifying and factoring out \(x\), we are left with \(x(x^{2}-3x-4)\), which is a simpler expression to work with and represents the essence of common factor identification.

Quadratic Equation Factoring

Quadratic equation factoring is the process used to break down a quadratic expression into multiple binomial factors. Once we have an expression like \(x^2 - 3x - 4\), factoring further requires finding two numbers that multiply to give the constant term (-4 in this instance) and add up to the coefficient of the \(x\) term (-3 here).

For the equation given in the example, we're looking for two factors of -4 that add up to -3. We determine that -4 and +1 fulfill this criteria because \(-4 \times 1 = -4\) and \(-4 + 1 = -3\). Consequently, our quadratic expression can be factored into \((x - 4)(x + 1)\).

Why Does This Work?

The act of factoring reverses the 'FOIL' process (First, Outer, Inner, Last), which is typically used to expand binomials. Through this strategy, we can simplify the quadratic to more basic expressions that make finding solutions or further simplifications easier.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent a specific quantity. In the problem \(x^{3}-3x^{2}-4x\), we're working with a polynomial, a type of algebraic expression.

Once we have identified common factors and factored quadratic equations, we combine them to express the original polynomial in its factored form. Bringing together the common factor \(x\) that we factored out previously and the factors of the quadratic, \((x - 4)(x + 1)\), we reach the finalized factored form \(x(x - 4)(x + 1)\). This transformation is crucial for solving equations and simplifying expressions in algebra, as it often reveals the roots or solutions to the equation—values of \(x\) that make the equation equal to zero.

Application in Solving Equations

Exploring algebraic expressions through problems like these helps to build a deeper understanding of the structures and patterns in algebra, making it easier for students to tackle even more complex problems efficiently and with confidence.