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An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as add, subtract, multiply, and divide). A key aspect of algebraic expressions is the presence of variables, which are symbols used to represent unspecified numbers or values.

An expression can be simple, with just one variable and no operations, such as 'a', or it could be more complex, such as the trinomial in our exercise \(4 y^{2}+y-7\). When working with algebraic expressions, particularly in the context of multiplication, one useful tool is the distributive property. This property states that a term multiplied by a sum or difference (like \(a(b + c)\)) is the same as the sum or difference of the individual products (\(ab + ac\)).

Understanding the distributive property is crucial because it allows us to simplify algebraic expressions by expanding and multiplying them out, exactly as demonstrated in our original exercise.

When it comes to polynomial multiplication, we're dealing with a scenario that involves multiplying polynomials together to get a new polynomial. A 'polynomial' is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Examples of polynomials include monomials (like \(3x\)), binomials (such as \(2y - 1\)), and trinomials (like \(4 y^{2}+y-7\)).

For our exercise, the process of polynomial multiplication involves using the distributive property to individually multiply each term of one polynomial by every term of the other polynomial. This is often referred to as the FOIL method when multiplying a binomial by a binomial—representing 'First', 'Outer', 'Inner', 'Last'. Although in this case, since we have a trinomial and a binomial, we systematically distribute each term of the binomial across all terms of the trinomial. The resultant terms are then simplified if possible, as seen in the step-by-step solution of the original exercise.

Once we have expanded the algebraic expressions through polynomial multiplication, the next step is combining like terms. Like terms are terms in an expression that have the same variables raised to the same power, even if they have different coefficients. In other words, they are terms that are 'alike' in their variable parts.

For instance, in the expression \(8 y^{3} - 4 y^{2} + 2 y^{2} - y - 14 y + 7\), the terms \-4 y^{2}\ and \(2 y^{2}\) are like terms because they both contain \(y^{2}\).

Why do we combine like terms?

It simplifies the expression, making it easier to understand and work with. The process involves adding or subtracting the coefficients of these terms while keeping the variable part the same.

In the original example, after combining like terms, we get \(8 y^{3} - 2 y^{2} - 15 y + 7\), a simplified and more manageable expression that conveys the same mathematical meaning as the expanded form.