Problem 3 - College Algebra with Modeling and Visualization

Question

Factor out the greatest common factor:. \(2 x^{3}-5 x\)

Step-by-Step Solution

Answer

The factored expression is \(x(2x^2 - 5)\).

Step 1: Identify Common Factors

Examine each term in the expression to determine what they have in common. For the given expression \(2x^3 - 5x\), note that both terms have an \(x\).

Step 2: Determine the Greatest Common Factor

List the factors of both coefficients: for 2 and 5, the greatest common factor is 1. For the variables, \(x^3\) and \(x\), the greatest common factor is \(x\). Therefore, the GCF of the whole expression is \(x\).

Step 3: Divide Each Term by the GCF

Divide each term by the greatest common factor, \(x\): \(\frac{2x^3}{x} = 2x^2\) and \(\frac{-5x}{x} = -5\).

Step 4: Write the Factored Expression

Combine the results from Step 3 into the factored expression: \(x(2x^2 - 5)\).

Key Concepts

Greatest Common Factor

One of the key steps in factoring polynomials is identifying the greatest common factor (GCF). This is the largest factor that divides each term in the expression without leaving a remainder. To find the GCF:

• First, look at the numerical coefficients of the terms. Break them down into their prime factors if necessary.
• Determine the highest power of common variables. For instance, in the expression \(2x^3 - 5x\), the common variable is \(x\).

In this case, since the coefficients 2 and 5 have no common factor other than 1, the GCF for the whole polynomial is simply the variable \(x\). Thus, finding the GCF helps simplify the polynomial into a more easily workable form, making further algebraic manipulations more manageable.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can represent numbers, variables, and operations. An expression like \(2x^3 - 5x\) includes terms that are either multiplied by each other or combined through addition and subtraction. Understanding algebraic expressions involves:

• Recognizing terms: A term consists of numbers and/or variables multiplied together, such as \(2x^3\) and \(-5x\) in our example.
• Coefficients and variables: The coefficient is the numeric part (2 and -5), and the variable is the symbol (\(x\)) representing unknown quantities.
• Operations: The terms are combined using addition or subtraction.

By breaking down expressions into their components, you gain insight into how to manipulate them, such as through operations like factoring, which rewrites them in a simpler or different form.

Factoring Techniques

Factoring polynomials involves several techniques, with the primary goal of simplifying the expression or solving equations. The technique applied depends on the structure of the polynomial. For expressions like \(2x^3 - 5x\), factoring out the greatest common factor is a straightforward approach. Here's how factoring works:

• Identify the GCF of all terms, as we found \(x\) for our example.
• Factor this GCF out of the polynomial, which involves dividing each term by the GCF and rewriting the polynomial as a product of this common factor and the simplified expression.
• Combine the results to form the final factored expression. For \(2x^3 - 5x\), factoring gives us \(x(2x^2 - 5)\).

By mastering various factoring techniques, you can handle more complex polynomials and solve polynomial equations efficiently.