Problem 6 - College Algebra with Modeling and Visualization

Question

Factor out the greatest common factor:. \(-5 x^{3}+x^{2}-4 x\)

Step-by-Step Solution

Answer

The greatest common factor is \(x\); expression simplifies to \(x(-5x^2 + x - 4)\).

Step 1: Identify the terms

The given polynomial expression is \(-5x^3 + x^2 - 4x\).We have three terms: \(-5x^3\), \(x^2\) and \(-4x\).

Step 2: Find the common factors

List the factors for each term:1. \(-5x^3\): Factors are \(-5, x, x, x\).2. \(x^2\): Factors are \(x, x\).3. \(-4x\): Factors are \(-4, x\).The common factor is \(x\), as it is present in each term.

Step 3: Factor out the greatest common factor (GCF)

Extract the \(x\) from each term:\(-5x^3 + x^2 - 4x = x(-5x^2 + x - 4)\).This simplification shows the expression factored by its greatest common factor.

Step 4: Verify the result

Distribute the factored term back into the expression to ensure correctness:\(x(-5x^2 + x - 4) = -5x^3 + x^2 - 4x\).The distribution matches the original polynomial, confirming the factorization.

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) plays a crucial role when simplifying expressions, especially polynomials. It's the largest factor shared by all terms in a polynomial.
Think of it as the biggest number or expression that can evenly divide all terms inside a polynomial. In our example, we dealt with the polynomial \(-5x^3 + x^2 - 4x\), where each term has a shared factor of \(x\).
Finding the GCF involves several steps:

• List out the factors of each term in the polynomial.
• Identify any factors that appear in every term.
• Select the largest of these shared factors, which becomes the GCF.

Identifying this can simplify complex expressions, making algebraic problems more manageable.

Polynomial Expressions

Polynomial expressions are a combination of terms made up of constants, variables, and exponents. These are fundamental in algebra and appear in various forms and operations.
For instance, our given expression \(-5x^3 + x^2 - 4x\) is a polynomial consisting of three terms: \(-5x^3\), \(x^2\), and \(-4x\).
The characteristics of polynomial expressions include:

• They consist of one or more terms added or subtracted together.
• Each term is a product that may include coefficients and variables raised to a power.
• The degree of a polynomial is determined by the highest power of the variable present in the expression.

Understanding polynomials is essential since they form the basis for solving equations and graphing functions.

Algebraic Factoring

Algebraic factoring involves breaking down complex expressions into simpler, "multiply-able" components. This skill is vital for solving equations, simplifying expressions, and manipulating algebraic functions more easily.
In particular, factoring a polynomial like \(-5x^3 + x^2 - 4x\) begins with finding the GCF and then expressing the polynomial as a product of the GCF and another polynomial.
To achieve this:

• After identifying the GCF, divide each term by this factor.
• Place the GCF outside of a parenthesis and write the simplified polynomial inside.
• Verify your work by distributing to ensure the original expression is obtained.

Such methods allow for further operations, be it solving for variables or integrating into larger algebraic functions.