Problem 19 - Elementary and Intermediate Algebra

Question

Factor each polynomial. $$ 2 x^{2}-6 x $$

Step-by-Step Solution

Answer

The polynomial factors to \(2x(x-3)\).

Step 1: Identify Common Factors

First, look at the polynomial and identify any common factors in each term. Both terms, \(2x^2\) and \(-6x\), have \(2x\) as a common factor. Rewrite the polynomial highlighting the common factor:\[2x(x - 3)\]

Step 2: Factor Out the Common Term

Now, factor out the common \(2x\) from each term in the polynomial. This will give us:\[2x(x - 3)\]

Step 3: Verify the Factoring

Multiply the factors \(2x\) and \(x - 3\) back to ensure they give the original polynomial:\[2x \cdot x - 2x \cdot 3 = 2x^2 - 6x\]The multiplication results in the original polynomial, confirming the factoring is correct.

Key Concepts

Common Factors

To understand factoring polynomials, identifying **common factors** is crucial. Imagine common factors as shared parts in different terms of a polynomial. A polynomial is a collection of terms added or subtracted together, and each term may have coefficients and variables. Finding common factors is like looking for what you can "take out" or simplify from these terms.

Let's simplify: If you have two numbers, say 12 and 8, the number 4 is a common factor because you can divide both 12 and 8 by it without leaving a remainder. Similarly, in a polynomial, a common factor exists if it is part of each term. For example, in terms like \(2x^2\) and \(-6x\), you have to inspect both the numerical and variable parts to find what they share.

In our example, both terms share the common factor \(2x\), which can be factored out to simplify expressions. Recognizing common factors is a foundational skill for polynomial simplification and sets the stage for more complex algebraic operations.

Polynomial Expressions

**Polynomial expressions** are constructed from variables and coefficients by applying addition, subtraction, multiplication, and non-negative integer exponents. Think of a polynomial as a string made up of terms. Each term is a combination of a number (coefficient) and a variable (like \(x\)) that is raised to some power.

For example, in the polynomial \(2x^2 - 6x\), \(2x^2\) is one term and \(-6x\) is another. Here, \(2\) and \(-6\) are coefficients, and \(x\) is the variable. The power of each variable in a polynomial tells you how the expression behaves - higher powers mean more complexity.

Polynomial expressions can include:

• Constant terms, like 3 or -4, which don’t have variables.
• Linear terms, like \(x\), where the variable is raised to the 1st power.
• Quadratic terms, like \(x^2\), where the variable is squared.

Knowing how to navigate and manipulate these expressions is key in algebra, calculus, and beyond.

Algebraic Factoring Steps

**Algebraic factoring** involves breaking down a polynomial into a product of simpler expressions. The goal is to simplify the process of solving equations by making them easier to handle.

At the heart of the factoring process, especially in algebra, is recognizing patterns and using them strategically. Here's a basic set of steps you can follow:

• **Identify common factors:** Look at each term in the polynomial, and see if they share a factor that can be removed.
• **Factor out the common element:** Write the polynomial as a product of the common factor and a simpler polynomial.
• **Verify by expansion:** Multiply the factors back together. If you get the original polynomial, your factoring is correct.

In our exercise with the polynomial \(2x^2 - 6x\), we identified \(2x\) as the common factor, then rewrote the expression: \(2x(x - 3)\). By expanding, we verified that \(2x(x - 3) = 2x^2 - 6x\) is indeed the same as the original polynomial.

These steps pave the way for tackling more advanced forms of polynomials, such as quadratic and cubic expressions, by providing a repeatable method.