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When it comes to polynomial multiplication, understanding how to multiply coefficients is crucial. Coefficients are the numerical part of the terms in an algebraic expression. Let's consider an example to grasp this concept better.

Imagine you have the terms \(10x^2y\) and \(5xy\). To find the product, you first look at the numbers in front of the variables, which in this case are 10 and 5. These are your coefficients. Multiplying them together, \(10\times 5 = 50\), gives you the new coefficient for the product. This step is quite straightforward; just remember to handle the coefficients before dealing with the variables.

Once you've handled the coefficients, the next step is to apply the laws of exponents correctly. These laws govern how we manipulate the powers of numbers and variables. The most relevant law for polynomial multiplication is the product of powers rule, which states that when you multiply two terms with the same base, you can add their exponents.

Take the previous example, where we multiply \(x^2\) by \(x\). Here, \(x\) actually has an exponent of 1, which is usually not written since anything raised to the power of 1 is itself. For multiplication, you would add exponents: \(2 + 1 = 3\), which gives \(x^3\). It's essential to remember that this rule only applies when the base is the same; otherwise, the terms cannot be combined this way.

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \(x\) or \(y\)), and operations like addition or multiplication. To fluently work with algebraic expressions, especially when multiplying them, it's crucial to perform each operation step by step. After dealing with coefficients and exponents, you must then combine like terms, which are terms that have the same variables raised to the same power.

In the earlier example of multiplying \(10x^2y\) and \(5xy\), after you've found the new coefficient (\(50\)) and worked out the variable part (\(x^3y^2\)), you combine them to form the final product, \(50x^3y^2\). This process of combining the numerical part and the variable part applies to all terms in a multiplication problem involving polynomials, enabling you to simplify expressions and find their products correctly.