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The greatest common factor (GCF) is the largest number or algebraic expression that divides each term in a polynomial without a remainder. When factoring by grouping, identifying the GCF in each group is essential.
For example, in the expression \(mn - 6m - 4n + 24\), we first group the terms into pairs \(mn - 6m and -4n + 24\).
Next, we find and factor out the GCF from each pair:

• From \(mn - 6m\), the GCF is \(m\) resulting in \(m(n - 6)\).
• From \(-4n + 24\), the GCF is \(-4\) resulting in \(-4(n - 6)\).

Thus, by factoring out the GCF, we simplify the polynomial and make it easier to combine or further factor.

Polynomial factoring involves expressing a polynomial as a product of simpler polynomials. This method is vital for solving polynomial equations, simplifying complex expressions, and finding roots.
Factoring by grouping is a technique used when a polynomial has four or more terms. It breaks down the problem into manageable parts.
Using our example \(mn - 6m - 4n + 24\), we grouped it into \(mn - 6m\) and \(-4n + 24\).
Factoring out the GCF from these groups gives us \(m(n - 6)\) and \(-4(n - 6)\).
We then combine these factored expressions and factor out the common binomial factor \(n - 6\): \(m(n - 6) - 4(n - 6) = (m - 4)(n - 6)\). This final factored form is simpler and ready for further manipulation or solving.

Algebraic expressions consist of numbers, variables, and operations (addition, subtraction, multiplication, division). Learning to factor and manipulate these expressions helps in simplifying and solving equations.
For example, consider the expression \(mn - 6m - 4n + 24\). It includes variables \(m\) and \(n\).
By factoring by grouping, we first group the terms and then find the GCF for each group:\