Terms, Coefficients & Like Terms - Sixth Grade Mathematics
Dissecting the interior anatomy of an algebraic expression identifies terms, coefficients, constants, and like terms, establishing the structural foundation for simplifying algebra.
The Anatomy of an Expression
An algebraic expression is constructed from distinct building blocks separated by addition and subtraction signs.
Expression: 5x^2 - 7x + 12
|__| |_| |__|
Term 1 Term 2 Term 3
- Total Terms: 3 terms (Trinomial)
- Coefficients: 5 (for x^2) and -7 (for x)
- Constant: 12
- Variable: x
Terms: The Units of an Expression
A term is a single number, a single variable, or the product of numbers and variables. Terms are separated from each other by addition (+) or subtraction (-) signs. The expression 4x + 3y - 9 contains three terms: 4x, 3y, and -9. The sign in front of a term belongs to that term! The third term is negative 9 (-9), not positive 9.
Coefficients: The Numerical Multipliers
A coefficient is the numerical factor that multiplies a variable in a term. In the term 8x, the coefficient is 8. In the term -5y, the coefficient is -5. In the term x/4, the coefficient is 1/4 (since x/4 = 1/4 * x).
The Invisible Coefficients: One and Negative One
When a variable appears alone without any visible number in front of it, its coefficient is understood to be 1: The term x means 1x. The coefficient is 1. The term -x means -1x. The coefficient is -1. Recognizing these invisible coefficients is vital when combining like terms.
Constants: Fixed Numerical Terms
A constant is a term that contains only a fixed number, with no variables attached. In the expression 6n + 15, the number 15 is a constant because its value remains 15 regardless of what number n represents.
Like Terms versus Unlike Terms
Like terms are terms that possess the EXACT SAME variables raised to the EXACT SAME exponents. Their coefficients may differ, but their variable anatomy must be identical.
+-------------------------------------------------------------------------+
| LIKE TERMS VERSUS UNLIKE TERMS |
+-------------------+-----------------------+-----------------------------+
| Pair of Terms | Classification | Reason |
+-------------------+-----------------------+-----------------------------+
| 4x and 9x | LIKE TERMS | Identical variable: x^1 |
+-------------------+-----------------------+-----------------------------+
| 7a^2 and -2a^2 | LIKE TERMS | Identical variable & power: |
| | | a raised to power 2 |
+-------------------+-----------------------+-----------------------------+
| 5x and 5y | UNLIKE TERMS | Different variables (x vs y)|
+-------------------+-----------------------+-----------------------------+
| 3x^2 and 3x | UNLIKE TERMS | Different exponents! |
| | | (x^2 vs x^1 are NOT like!) |
+-------------------+-----------------------+-----------------------------+
| 8 and -14 | LIKE TERMS | Both are pure constants |
+-------------------+-----------------------+-----------------------------+
Exercises
Exercise 1. Identify the number of terms, list each term, identify each coefficient, and name the constant in the expression: 6x^2 - 4x + 9y - 15.
Exercise 2. State the coefficient of the variable in each term: (a) 12m; (b) -y; (c) w; (d) -7k; (e) p/3.
Exercise 3. Group the following terms into sets of like terms: 4x, 7y, -2x, 5, x^2, -3y, 11, 4x^2.
Solutions
Solution to Exercise 1. In 6x^2 - 4x + 9y - 15: there are 4 terms. The terms are 6x^2, -4x, 9y, and -15. The coefficients are 6 (for x^2), -4 (for x), and 9 (for y). The constant term is -15.
Solution to Exercise 2. For (a) 12m: coefficient is 12. For (b) -y: coefficient is -1. For (c) w: coefficient is 1. For (d) -7k: coefficient is -7. For (e) p/3: coefficient is 1/3.
Solution to Exercise 3. Grouping by identical variables and powers: Set 1 (x-terms): { 4x, -2x } Set 2 (y-terms): { 7y, -3y } Set 3 (x^2 terms): { x^2, 4x^2 } Set 4 (constants): { 5, 11 }