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When factoring polynomials, identifying the Greatest Common Factor (GCF) is the essential first step. This refers to the largest factor that is shared by all terms within the polynomial expression.
For coefficients like 15, 33, and 6, find the GCF by determining the largest number that divides each of these coefficients. Here, the GCF is 3.
Additionally, consider the variables. Examine each term to determine the smallest power of each variable that appears across all terms. In our exercise, the lowest power of the variable \(a\) is \(a^2\), and for \(b\), it is \(b\). Combining these, we find that the GCF for the entire polynomial is \(3a^2b\). This is the first step towards simplification.

Polynomial expressions are mathematical phrases that can include variables, coefficients, and exponents. They are typically written in a form where terms are added or subtracted.
Each term in a polynomial is composed of a coefficient, a variable (or multiple variables), and an exponent. For instance, in the term \(15a^4b\), the number 15 is the coefficient, \(a^4\) indicates that \(a\) is raised to the fourth power, and \(b\) is a single variable.
Understanding the structure of polynomial expressions is crucial for effectively working with them, particularly when factoring. In our problem, the polynomial is \(15a^{4}b - 33a^{3}b^{2} + 6a^{2}b^{3}\). Notice the like terms grouped by the presence of variables \(a\) and \(b\) raised to various powers. Recognizing this allows us to streamline the expression by factoring.

After factoring a polynomial expression, it’s important to confirm the results by expanding the expression again. This process, known as expansion, ensures that the factorization was done correctly.
To verify, distribute the factored terms back across the expression. For the factorization \(3a^2b(5a^2 - 11ab + 2b^2)\), distribute \(3a^2b\) to each term within the parentheses:

• Expanding \(3a^2b \times 5a^2\) yields \(15a^4b\).
• Expanding \(3a^2b \times (-11ab)\) yields \(-33a^3b^2\).
• Finally, expanding \(3a^2b \times 2b^2\) yields \(6a^2b^3\).

By matching these results with the original polynomial, \(15a^{4}b - 33a^3b^{2} + 6a^2b^{3}\), we confirm that the factorization was performed accurately. This process ensures that your factorization steps are both logical and mathematically sound.