91Ó°ÊÓ

To fully grasp factoring polynomials, start by understanding that a polynomial is an expression made of variables, exponents, and coefficients. Factoring is the process of breaking down the polynomial into simpler terms (or factors) that, when multiplied together, give the original polynomial.

In the exercise given, you see the polynomial: \(12a^{2} - 4bc + 16ac - 3ab\). This is a specific case where we use a method called 'factoring by grouping'.

This method involves breaking the polynomial into groups where each group can be factored individually. This is a helpful and practical way to simplify complex expressions and solve equations.

Polynomials fall under a broad category called algebraic expressions. These are mathematical phrases that include numbers, variables, and operation symbols.

For example, in our exercise, \(12a^{2} - 4bc + 16ac - 3ab\) is an algebraic expression. Each term in this expression is a combination of numbers (called coefficients) and variables (like \(a\), \(b\), and \(c\)).

When we factor by grouping, we're essentially looking at sub-expressions within the polynomial. We group these sub-expressions in ways that make it easier to see common factors and simplify the entire expression.
In this process:

• Divide the terms into smaller groups.
• Find a common factor in each group.
• Factor each group separately before combining them back into a single, factorable format.

Identifying common factors is a crucial step in factoring polynomials by grouping. A common factor is a number or variable that is present in each term of a group.

In the first group of our exercise, \(12a^2 - 4bc\), the common factor is \(4a\). This means that both terms can be divided by \(4a\). So, when we factor out \(4a\), we get \(4a(3a - bc)\).

For the second group, \(16ac - 3ab\), finding a common factor is slightly trickier. Here, we can factor out \(-b\), yielding \(-b(16a - 3a)\). When we simplify \(16a - 3a\), we get \(13a\), making the group \(-b(13a)\).

Recognizing and using common factors simplifies polynomials and makes expressions easier to work with. This skill is vital in algebra and helps build a foundation for more complex math topics such as solving equations and calculus.