1. Introduction
Algebraic expressions are made up of smaller parts. These parts are called terms. Each term may contain numbers and letters multiplied together. The number part of a term is called the coefficient.
In this topic, we will learn how to identify terms and coefficients in different algebraic expressions. This will help you in addition, subtraction, and multiplication of expressions later.
2. Terms in an Algebraic Expression
Term: A term is a part of an algebraic expression that is separated by a plus (\(+\)) or minus (\(-\)) sign.
For example, in the expression \(5x + 3y - 7\):
- The terms are \(5x\), \(3y\), and \(-7\).
- They are separated by the signs \(+\) and \(-\).
2.1. Examples of Terms
Look at these expressions and their terms:
| Expression | Terms |
|---|---|
| \(4x + 5\) | \(4x\), \(5\) |
| \(3a - 2b + 7\) | \(3a\), \(-2b\), \(7\) |
| \(6m - 3n - 9\) | \(6m\), \(-3n\), \(-9\) |
2.2. Number of Terms
The number of terms in an expression is simply how many separate pieces it has.
- \(4x\) has 1 term.
- \(2x + 3\) has 2 terms.
- \(x + y + z\) has 3 terms.
3. Coefficients in a Term
Coefficient: The coefficient of a term is the number (or fixed value) that multiplies the variable.
For example, in the term \(5x\):
- \(5\) is the coefficient.
- \(x\) is the variable.
In the term \(-3y\):
- \(-3\) is the coefficient.
- \(y\) is the variable.
3.1. Numeric and Algebraic Coefficients
Sometimes, a coefficient can itself contain letters.
- Numeric coefficient: The number part of a term. In \(7xy\), the numeric coefficient is 7.
- Algebraic coefficient: The part containing letters that multiply a variable. In \(7xy\), the algebraic coefficient of \(x\) is \(7y\).
3.1.1. Examples Table
| Term | Variable | Numeric Coefficient |
|---|---|---|
| \(5x\) | \(x\) | 5 |
| \(-3y\) | \(y\) | -3 |
| \(8ab\) | \(ab\) | 8 |
| \(-2mn\) | \(mn\) | -2 |
4. Factors of a Term
Each term can be broken into factors. A factor is a quantity that is multiplied to get the term.
For example, in the term \(6xy\):
- Factors are \(6\), \(x\), and \(y\).
- We can write \(6xy = 2 \times 3 \times x \times y\).
4.1. Term, Factors, and Coefficient Together
| Term | Factors | Coefficient |
|---|---|---|
| \(4x\) | \(4, x\) | 4 |
| \(-5y\) | \(-5, y\) | -5 |
| \(3ab\) | \(3, a, b\) | 3 |
| \(-2mn\) | \(-2, m, n\) | -2 |
5. Identifying Terms and Coefficients in Expressions
To understand any expression, first split it into terms, then find the coefficient of each term.
Example 1: In \(7x + 4\):
- Terms: \(7x\), \(4\)
- Coefficient of \(x\): 7
- Coefficient of constant term: 4 (it has no variable)
Example 2: In \(3a - 5b + 2\):
- Terms: \(3a\), \(-5b\), \(2\)
- Coefficient of \(a\): 3
- Coefficient of \(b\): -5
- Constant term: 2
5.1. Common Mistakes
- Forgetting the sign when writing the coefficient. For example, in \(-4x\), the coefficient is -4, not 4.
- Counting each letter as a separate term instead of seeing the whole group (like \(3xy\) is one term, not three).
- Thinking constants do not have coefficients. A constant like 5 can be seen as \(5 = 5 \times 1\), so its coefficient is 5.
6. Quick Practice
Try these on your own:
- Write the terms and coefficients in \(4x + 7\).
- In \(3a - 2b + 5\), find the coefficient of \(a\) and \(b\).
- For the expression \(6m - 3n - 9\), list all the terms and their coefficients.
- In \(-2xy + 5x - 7\), what are the coefficients of \(xy\) and \(x\)?
7. Summary
- A term is a part of an expression separated by plus or minus signs.
- The coefficient is the number that multiplies the variable in a term.
- Coefficients can be positive or negative and may be hidden as 1 or -1.
- Each term can be broken into factors, including its coefficient and variables.
- Understanding terms and coefficients makes it easier to work with algebraic expressions.