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Simplification in algebra is the process of reducing an expression to its simplest form. This involves combining like terms and making any necessary arithmetic operations clearer. In our exercise, the expression is given as \(1ab + 21a + 1b\). The first task is to look for like terms. Like terms are terms that have the same variables raised to the same power, only differing in their coefficients.

However, in this expression, the terms \(ab, 21a,\) and \(b\) each contain different variables or combinations of them. They cannot be combined because they are not like terms. When there are no like terms to combine, the expression is already simplified to its lowest form, so our expression \(ab + 21a + b\) remains unchanged after simplification.

Variables are symbols in algebra that represent unknown quantities. They are often denoted by letters such as \(a\), \(b\), \(x\), \(y\), and so on. In our expression, we have two variables: \(a\) and \(b\). These variables can represent any number and are commonly used in algebra to generalize mathematical expressions.
Variables allow you to set up equations and expressions when not all quantities are known. For example, if you know the coefficient of \(a\) but not its value, the variable \(a\) stands in its place until you find out more information.
Understanding that variables are placeholders for numbers helps make sense of why certain terms cannot be combined unless they possess exactly the same variables in the same order.

Coefficients are the numerical parts of terms that multiply the variables. In the expression \(1ab + 21a + 1b\), the terms \(ab\), \(21a\), and \(b\) each have coefficients. Coefficients are important because they tell you "how many" of each term you have.
For instance, the expression \(21a\) means that you have 21 instances of whatever number \(a\) represents. Similarly, \(ab\) and \(b\) can be understood as 1(ab) and 1(b), showing that the coefficient is 1 even when it is not visible. Listing coefficients explicitly can clarify expressions, especially when deciding if and how terms can be combined during simplification.
Knowing the coefficient helps in performing operations on expressions, like multiplying or dividing terms, and in understanding the relative size or influence of different terms.