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Polynomial distribution is a fundamental process in algebra that involves multiplying a single term by each term inside a set of parentheses. In our exercise, we began by distributing the term outside the parentheses to simplify the expression. This means we took the term 5y and multiplied it with every term inside the parentheses. For example, distributing 5y to \((y^2 - 3)\), we get: \((5y \times y^2) - (5y \times 3)\). This step-by-step multiplication ensures each term inside the parentheses is individually handled, making it easier to manage complex expressions. When performing polynomial distribution, always check to multiply each term correctly to avoid any mistakes.

Combining like terms is a technique used to simplify algebraic expressions by merging terms that have the same variables raised to the same power. In our exercise, after distributing, we had: \((5y^3 - 15y)\) and \((y^3 - 3y^2)\). These terms need to be combined by identifying and adding or subtracting like terms.
A like term has the same variables with the same exponents. For instance, 5y^3 and y^3 are like terms, and they can be combined to become 6y^3. Likewise, different sets of terms such as \(-15y\) and \(-3y^2\) are managed separately because they are not like terms. Recognizing and combining like terms properly ensures the expression is simplified accurately.

Algebraic simplification is the process of making an algebraic expression as simple as possible by performing operations like distribution and combining like terms. This process is crucial in solving equations efficiently.
In our example, after proper distribution and combining like terms, the initial complex expression \((5y(y^2-3) + y^2(y-3))\) was simplified step-by-step, leading to \((6y^3 - 3y^2 - 15y)\). Each step, from distributing terms to grouping similar terms, reduces the complexity and gets us closer to the simplest form. Remember, always keep your work organized and check each step to avoid mistakes. By breaking down the process, algebraic simplification becomes manageable and straightforward.