91Ó°ÊÓ

When studying algebra, you're bound to encounter mathematical coefficients. These are the numerical factors that multiply the variables in an expression. For instance, in the term \( -5.16xy \), \( -5.16 \) is the coefficient of the variable \( xy \). It's important to distinguish them because they directly influence the value of the variable expression.

Here's a quick tip: keep track of the sign of the coefficient. If it's negative, as in our example, it will flip the sign of the product every time you distribute it across an addition or subtraction within the parentheses. This is essential when applying the distributive property to ensure you're combining terms correctly.

The rules of exponents are fundamental when working with polynomials. These rules dictate how to handle powers when you multiply or divide variables. In our exercise, we note that multiplying variables of the same base requires adding their exponents, such as \( x^a \cdot x^b = x^{a+b} \).

For example, if you have \( x^2 \cdot x \), according to the rules of exponents, you would add the exponents 2 and 1 (the understood exponent of \( x \) without an exponent) to get \( x^3 \). It’s like saying you have a group of \( x \) factors being multiplied a certain number of times, and the exponent simply tells you how many there are in total. These rules keep the algebraic manipulations tidy and systematic.

The polynomial expansion occurs when you multiply out terms of a polynomial that are being added or subtracted, typically within parentheses. The goal is to remove the parentheses to write the polynomial as a sum of terms. In our problem, we apply the distributive property to expand the polynomial \( -5.16xy(1.23x^2y - 5.83xy^2 + 4.27x^2y^2 - 2.94xy) \).

Each term inside the parenthesis gets multiplied by \( -5.16xy \), and this is where being meticulous with coefficients and exponent rules pays off. This process results in a longer expression where no parentheses remain, and like terms can then be combined if necessary. It’s a bit like unpacking a suitcase where items represent terms, ensuring that everything is visible and has its place.