91Ó°ÊÓ

When we multiply polynomials, we are essentially combining like terms after distributing each term of one polynomial by every term of another. In our exercise, we encounter a special case known as the square of a binomial.

Consider the product of two binomials \( (a + b)(c + d) \). The process involves multiplying each term in the first binomial by each term in the second binomial, commonly referred to as the FOIL (First, Outer, Inner, Last) method. When squaring a binomial—similar to our exercise where we have \( (x^2 y^2 - 5)^2 \)—we are simply applying this FOIL method as well. However, since both binomials are identical, it's like distributing \( a + b \) onto itself: \( a^2 + 2ab + b^2 \).

This process embeds the essential strategy of polynomial multiplication: aligning like terms to combine them, leading to a simplification of the expression.

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \( x \) or \( y \) in our exercise), and operators (such as addition, subtraction, multiplication, and division). The goal when working with algebraic expressions is to manipulate these expressions using algebraic rules in order to simplify or transform them.

In the context of our exercise \( (x^{2} y^{2}-5)^{2} \) we are working with a binomial squared which itself is an algebraic expression. The expression is composed of variables \( x^2 y^2 \) and constants like -5. By following specific algebraic principles—like the distributive property and laws of exponents—we are able to expand and simplify the expression to reveal its simplified form without changing its value.

Simplifying expressions is a primary goal in algebra and involves reducing an expression to its most basic form while maintaining its original value. This process may involve combining like terms, factoring, and employing various algebraic rules.

In our solved problem, after the binomial expansion, we simplify the expression \( (x^{2} y^{2})^2 - 2(x^{2} y^{2})(5) + 5^2 \) by first calculating the powers and products to get \( x^{4} y^{4} - 10x^{2} y^{2} + 25 \). There are no like terms to combine further, which means the expression is fully simplified. The last part, often overlooked, is to check if the terms themselves can be simplified or factored further, but in this case, they cannot. The focus during simplification should always be on careful arithmetic and adherence to algebraic operations to ensure accuracy.