In Exercises 67–82, find each product. (7x^2 y+1)(2x^2 y−3)

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves distributing each term in one polynomial to every term in another polynomial. This process is often referred to as the distributive property, where you multiply coefficients and add the exponents of like bases. For example, in the expression (7x^2 y + 1)(2x^2 y - 3), you would multiply 7x^2 y by both 2x^2 y and -3, and then do the same for the constant term 1.

Combining Like Terms

After multiplying polynomials, the next step is to combine like terms, which are terms that have the same variable raised to the same power. This simplification is crucial for expressing the final result in its simplest form. For instance, if the multiplication yields terms like 14x^4 y^2 and -21x^2 y, you would keep these separate as they are not like terms.

Exponents and Their Rules

Understanding exponents is essential in polynomial operations, as they dictate how to handle the multiplication of variables. The rules of exponents state that when multiplying like bases, you add the exponents (e.g., x^a * x^b = x^(a+b)). This concept is particularly important when dealing with terms that include variables raised to powers, ensuring accurate calculations during the multiplication process.