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The greatest common factor, often abbreviated as GCF, is a key tool in factoring algebraic expressions. It refers to the largest factor that divides each term in a set of terms without a remainder. By identifying the GCF, you can simplify expressions and make further algebraic operations easier.
In the expression we dealt with, which was given as \(7x^3 - 14x\), the task was to find the largest factor common to both terms. Let's break it down:

• Identify the numerical part: For 7 and -14, the greatest common number is 7.
• Identify the variable part: Both terms have a factor of \(x\) with the lowest power, which is \(x^1\).

Hence, the GCF here is \(7x\). Once you have identified the GCF, you can "factor it out," which means you essentially divide each term by this factor. It simplifies the expression by revealing what's left inside the parentheses, as seen in the second step of the solution.

Algebraic expressions involve numbers, variables, and operations combined in a meaningful way to represent a mathematical relationship or concept. They play a vital role in various branches of mathematics.
In the context of our exercise, you were presented with \(7x^3 - 14x\). This expression includes terms:

• Numerical Coefficients: The numbers 7 and -14 are coefficients of the terms.
• Variables: The letter \(x\) represents the variables, raised to different powers.

The goal was to express this algebraic expression in its simplest form by factoring out common factors, thus making it easier to work with or to solve related equations. To handle such expressions efficiently, it's crucial to understand how terms interact and how common factors can simplify the expression.

Polynomials are a specific type of algebraic expression where multiple terms are summed together. Each term in a polynomial consists of a coefficient (a number), a variable (like \(x\)), and an exponent indicating the power to which the variable is raised.
The expression in our task, \(7x^3 - 14x\), is a polynomial because it has more than one term and involves powers of a variable. Here are a few characteristics of polynomials demonstrated by the expression:

• Terms: The expression has two terms, \(7x^3\) and \(-14x\).
• Degrees: The degree of the polynomial is determined by the highest power of the variable, which is 3 in our case (from \(7x^3\)).
• Simplification: By factoring, polynomials can often be rewritten in simpler forms for easier manipulation.

Understanding polynomials' structure and function is fundamental in algebra as it helps in solving equations and understanding the relationships between variables in more complex forms.