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Polynomial simplification is all about making expressions easier to work with. When we simplify polynomials, we reduce them to their simplest form. This can involve combining like terms, removing parentheses, and performing arithmetic operations. The goal is a more compact and manageable expression.
Let's break this down with our exercise. We started with a polynomial:\[3 x^{2}+6 x y+3 y^{2}-5 x^{2}-10 x y\]
Our job is to simplify this by combining like terms. The process of simplification makes sure that polynomials are easier to understand and work with in mathematics. This can be crucial for solving equations and understanding the relationships between different mathematical expressions.

Like terms are terms in an algebraic expression that have the same variables raised to the same power. To simplify an expression, you combine like terms by adding or subtracting their coefficients. This process helps to consolidate the expression into a simpler form.
In our exercise, the terms involving \(x^2\) and \(xy\) are like terms. We identified:

• \(3 x^2\) and \(-5 x^2\)
• \(6 xy\) and \(-10 xy\)

These terms have the same variable parts, which means they can be combined. To combine them, we add or subtract their coefficients:
\[3 x^2 - 5 x^2 = -2 x^2\]
\[6 xy - 10 xy = -4 xy\]
This resulted in the simpler form of our original polynomial expression.Recognizing and combining like terms is a fundamental skill in algebra that helps to manage and solve complex expressions more efficiently.

An algebraic expression is a combination of constants, variables, and arithmetic operations. These expressions represent mathematical relationships and quantities. For example, in our exercise we've worked with the algebraic expression:
\[3 x^{2}+6 x y+3 y^{2}-5 x^{2}-10 x y\]
This expression includes variables \(x\) and \(y\), and arithmetic operations like addition and subtraction. Simplifying algebraic expressions often involves combining like terms to create a more straightforward, compact form.

• Step 1: Identify like terms.
• Step 2: Combine like terms.
• Step 3: Rewrite the expression in simplest form.

In complex problems, understanding and simplifying algebraic expressions is a key skill. It leads to easier and more manageable equations, making it possible to find solutions quickly and accurately.
By mastering the combination of like terms and the simplification of algebraic expressions, you can solve increasingly challenging mathematical problems with confidence.