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Factoring polynomials is a fundamental skill in algebra that involves breaking down a polynomial into simpler 'factor' expressions that, when multiplied together, give back the original polynomial. This process can simplify seemingly complex expressions and make solving equations much easier.
For example, consider the polynomial expression in our problem: \(u^{2} - u + 6u - 6\).
Here, we were able to simplify and factor this polynomial by grouping terms and identifying common factors. This polynomial was first grouped into \((u^2 - u) + (6u - 6)\) and then factored further.
Factoring polynomials usually involves these key steps:

• Group similar terms.
• Identify and factor out the Greatest Common Factor (GCF).
• Factor out remaining common factors if possible.

Understanding these basics will greatly benefit you in solving more complex algebraic expressions.

The Greatest Common Factor (GCF) is the largest factor that two or more terms share. Identifying the GCF is an essential step in the process of factoring polynomials because it simplifies each term to its most reduced form, making the overall expression easier to work with.
In the example from our exercise, we identified the GCF in each group separately.
For \(u^2 - u\), the GCF is \(u\), and for \(6u - 6\), the GCF is \(6\). Factoring these out, we get \(u(u - 1) + 6(u - 1)\).
This simplification allows us to see further patterns or common factors, like the common binomial \(u - 1\) in our case, which was then factored out to give \( (u - 1)(u + 6) \). Recognizing and factoring out the GCF is a powerful technique in algebraic manipulation.

Algebraic expressions are mathematical phrases that can include numbers, variables (like \(u\) in our exercise), and operation symbols like plus (+) and minus (−). In our problem, the expression \(u^{2} - u + 6u - 6\) is an example of an algebraic expression.
Working with algebraic expressions often involves simplifying them or factoring them to find solutions to algebraic equations. Simplifying algebraic expressions can involve combining like terms, factoring, and other algebraic operations.
Being comfortable with these concepts is essential because it helps you understand how to manipulate and solve complex equations step-by-step. For instance, knowing how to combine like terms will allow you to see that \( -u + 6u \) is just \( 5u \), simplifying the expression significantly right from the start.
This understanding makes the broader picture of algebra clearer, helping you solve more complicated problems with confidence.