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Understanding the greatest common factor (GCF) is crucial when simplifying algebraic expressions. The GCF of two or more numbers is the highest number that can divide all the numbers without leaving a remainder. When dealing with algebraic expressions, the concept extends to include variables as well.

To find the GCF of algebraic terms, first list out the factors of the numerical coefficients. For example, the factors of 5 are 1 and 5, while for 20 they are 1, 2, 4, 5, 10, and 20. Next, identify any common factors of the variables by looking at the smallest exponent of that variable present in all terms. In the provided example, since the lowest power of the variable \(n\) is 1, the GCF includes \(n\) to the power of 1.

The final GCF is the product of the highest common numerical factor and the variable with the lowest power.

Polynomial factoring is a method used to break down complex algebraic expressions into simpler, multi-factorial components. It is akin to breaking down a number into its primes. When factoring polynomials, the goal is to find the product of simpler polynomials that, when multiplied together, will give the original polynomial.

After finding the GCF, which is the first and fundamental step, you would express the original polynomial as a product where the GCF is multiplied by another polynomial. If you can further factor the remaining polynomial, continue the process until the expression cannot be factored any more. For instance, the expression \(n^2 - 4\) in our exercise can be further factored into \((n + 2)(n - 2)\), using the difference of squares rule, but this step wasn't required in the given exercise.

Algebraic terms are the building blocks of algebraic expressions. Each term consists of numbers, variables, and possibly exponents. Terms are separated by a plus \(+\) or minus \(-\) sign within an expression.

In the given exercise \(5n^3 - 20n\), there are two terms: \(5n^3\) and \(-20n\). Each has a coefficient (5 and -20, respectively) and a variable part (\(n\) raised to some power). Understanding each term's components is key to simplifying and factoring expressions properly. The number in front of the variable is the coefficient, while the letter is the variable itself, which can have a power, indicating repeated multiplication of the variable.