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In this lesson, we will look at some different parts that make up a mathematical expression. Those parts are terms, factors, and coefficients. We will define each of these and see what they look like in various examples.

Parts of an Expression

Do you ever notice that the parts of a whole have their own individual labels and functions, but the whole couldn’t function properly without even one of them? For instance, consider a baseball team. When a baseball team is on a field, there are nine different players that take up different positions on the field. There is a pitcher, a catcher, a first, second, and third baseman, a shortstop, and a left, right and center fielder in the outfield.Each position on the field has its own label and its own function. When we put all these positions together, they work together to make up the whole, and if the team was missing a player at even one of these positions, the team’s performance would suffer.

Surprisingly, this concept also applies to mathematical expressions! A mathematical expression is a phrase that contains numbers, variables, symbols and operators grouped together using addition, subtraction, multiplication, and division. For example, 3x 2 + 7x is a mathematical expression. Each mathematical expression has different parts to it, and just like the players on a baseball team, these parts have different labels and different functions. Let’s take a look at some of these parts.

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Terms, Factors, and Coefficients

As we said, a mathematical expression contains numbers, variables, symbols, and operators connected by addition, subtraction, multiplication, and division. First, let’s consider the parts of the expression that are connected with addition and subtraction. These parts are called terms.

Consider our example 3x 2 + 7x. We see that 3x 2 and 7x are connected by addition. Therefore, in this mathematical expression, we call 3x 2 and 7x terms.

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Next, let’s talk about factors in a mathematical expression. In a mathematical expression, factors are parts of the expression that are connected by multiplication.

For instance, consider the mathematical expressions 5xy and (x – 2)(x + 5). In the first expression, we see we are multiplying 5 by x by y, so we would call 5, x, and y factors of the mathematical expression. Similarly, in the expression (x – 2)(x + 5), we are multiplying (x – 2) and (x + 5), so we call (x – 2) and (x + 5) factors of the expression.

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So far, so good, right? Let’s consider one more part of a mathematical expression.

The coefficients. A coefficient is a number that is multiplied by a variable in a mathematical expression. To illustrate this, consider the terms of our initial example:3x 2 + 7x. In each of the terms 3x 2 and 7x, x is a variable, and in both terms, we are multiplying a number by the variable. In the first term, 3x 2, 3 is being multiplied by the variable, so 3 is a coefficient.

In the second term, 7x, 7 is being multiplied by the variable, so 7 is a coefficient.

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We see that terms, factors, and coefficients make up mathematical expressions. These parts of an expression each have their own label and function.

The more we work with mathematical expressions, the more familiar we will become with these parts, so let’s take a look at some examples to really drive this concept home (baseball pun intended).

Examples

Suppose you throw a ball in the air, and the ball’s height from the ground can be modeled using the mathematical expression -16x 2 + 20x + 6.

We want to identify the terms and the coefficients in this expression. First, we consider the terms of the expressions, or the parts of the expression that are connected by addition or subtraction.

We see that the parts of the expression that are connected by addition or subtraction are -16x 2, 20x, and 6. Therefore, these are the terms of the expression.Now, let’s consider the coefficients, or the numbers that are multiplied by a variable in each of the terms. In the first term, -16x 2, -16 is being multiplied by the variable x, so -16 is a coefficient. In the second term, 20x, 20 is being multiplied by the variable x, so 20 is a coefficient.

In the last term, 6, there is no variable, so 6 is not called a coefficient. Actually, when we have a term that is just a number like this one, we call that number a constant, so 6 in this expression is a constant.

As it turns out, we can actually factor the mathematical expression -16x 2 + 20x + 6 to get (8x + 2)(-2x + 3). When we write the expression in its factored form, we have a new mathematical expression that has two factors. Can you identify the factors?If you said the factors are 8x + 2 and -2x + 3, then you are correct! In the expression (8x + 2)(-2x + 3), we are multiplying 8x + 2 and -2x + 3 together, so these two expressions are factors.

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Lesson Summary

A mathematical expression is an expression that contains numbers, variables, symbols, and operators connected with addition, subtraction, multiplication, and division. Each mathematical expression has different parts. Three of these parts are terms, factors, and coefficients.

The terms of a mathematical expression are parts of the expression that are connected with addition or subtraction. The factors in a mathematical expression are parts of the expression that are connected with multiplication. Lastly, the coefficients in a mathematical expression are the numbers that are multiplied by a variable.We use all of these different parts to create mathematical expressions.

This is very useful, because all of these parts come together to represent real world phenomena in a mathematical expression. It’s great that we are now familiar with each of these parts of the whole!

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