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A common factor is a term that appears in every part of an algebraic expression. To determine the common factors, look at each term. Here, we have \(2 \pi r h + 2 \pi r^{2}\). Notice that both terms share the numerical coefficient 2 and the variables \( \pi \) and \( r \).
If all terms in an algebraic expression share a common factor, you can simplify the expression by factoring out (or 'pulling out') this common factor. Factoring out means placing the common factor in front of a set of parentheses and dividing each original term by this common factor. So, from \(2 \pi r h + 2 \pi r^{2}\), we factor out \(2 \pi r\), resulting in \(2 \pi r (h + r)\). This step simplifies the expression. It's crucial to factor out the greatest common factor (GCF), which is the largest factor shared by all terms. In this case, the GCF is \(2 \pi r\).

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Here, \(2 \pi r h + 2 \pi r^{2}\) is an example of an algebraic expression. Each part of the expression separated by a plus or minus sign is called a 'term'.
Understanding the components of an algebraic expression helps in the process of factoring and simplifying it. When factoring an expression, you identify the common factors in each term. This reduces the complexity of the expression and can make further calculations easier.

• Variables: Symbols like \ r \ or \ h \ that represent unknown quantities.
• Coefficients: Numeric factors that multiply the variables, such as 2 in \(2 \pi r h\).
• Constants: Fixed values like \( \pi \), which in this case, is a mathematical constant.

By combining these elements in different ways, various algebraic expressions can be formed.

A prime polynomial is an expression that cannot be factored further using integer coefficients. It is already in its simplest form. In our example, after factoring out the common factors, we got \(2 \pi r (h + r)\).
The expression inside the parentheses, \( h + r \), cannot be simplified further because it doesn't share any common factors other than 1. Therefore, once you've factored out the greatest common factor and cannot further break down the resulting polynomial, you've reached a prime polynomial.
In summary, if after factoring out all common factors, you cannot find any further factors within the polynomial, then the polynomial is a prime polynomial.