What Is a Term in Algebra?
The Anatomy of an Algebraic Term
A term in its most common form is a product of a number and one or more variables raised to powers. Think of it as a package. This package has two main parts:
| Part Name | Description | Example in $-5x^2y$ |
|---|---|---|
| Coefficient[1] | The numerical factor of the term. It includes the sign (+ or −). If no number is written, the coefficient is $1$ or $-1$. | $-5$ |
| Variable Part | The letter(s) that represent unknown values, often raised to an exponent[2]. | $x^2y$ (which means $x \cdot x \cdot y$) |
| Degree[3] | The sum of the exponents of all variables in the term. For a constant term (just a number), the degree is $0$. | The exponent on $x$ is $2$, on $y$ is $1$ (since $y = y^1$). Degree = $2 + 1 = 3$. |
Types of Terms: From Constants to Monomials
Not all terms look the same. They are categorized based on what they contain.
Constant Term: A term that is only a number, with no variable part. In the expression $4x^2 + 9$, the number $9$ is a constant term. Its value never changes.
Variable Term: A term that contains at least one variable. $4x^2$, $-7ab$, and $y$ are all variable terms.
Most single terms you encounter in basic algebra are also called monomials. A monomial is a single term that is a product of a constant and variables with non-negative integer exponents. $5$, $-3t$, and $\frac{1}{2}m^2n$ are all monomials. Expressions like $x^{-2}$ or $\frac{3}{y}$ are not monomials because the exponents are negative (which is like division by a variable).
The Power of Like Terms: Combining and Simplifying
The real utility of understanding terms comes when you start to combine like terms. This is one of the most fundamental skills in algebra.
Like Terms: Terms that have the exact same variable part (the same letters each raised to the same powers). Only the coefficients can be different.
| Expression | Like Terms? | Reason |
|---|---|---|
| $3x$ and $-8x$ | Yes | Both have the variable part $x$. |
| $5y^2$ and $2y$ | No | The exponents of $y$ are different ($2$ vs. $1$). |
| $4ab$ and $-ba$ | Yes | $ba$ is the same as $ab$ (multiplication is commutative). |
| $7$ and $-3$ | Yes | Both are constant terms (degree 0). They are like terms. |
To combine like terms, you simply add or subtract their coefficients while keeping the variable part unchanged. It's like saying "3 apples + 5 apples = 8 apples." The "apple" (the variable part) stays the same.
Example: Simplify $2x + 5y - x + 3y$.
- Identify like terms: Terms with $x$: $2x$ and $-x$. Terms with $y$: $5y$ and $3y$.
- Combine their coefficients: $(2 - 1)x = 1x$ (usually written as just $x$). $(5 + 3)y = 8y$.
- The simplified expression is $x + 8y$.
Working with Terms in Equations and Formulas
Terms are not just passive parts of expressions; they are actively manipulated when solving equations and using formulas. The principle of "doing the same thing to both sides" of an equation often involves moving terms from one side to the other by adding or subtracting them.
Example Problem: Solve for $n$: $3n + 10 = n - 2$.
Step-by-Step Solution:
- The equation has terms on both sides: $3n$, $10$, $n$, and $-2$.
- Goal: Get all terms containing $n$ on one side, and constant terms on the other. Subtract $n$ from both sides: $3n - n + 10 = n - n - 2$. This simplifies to $2n + 10 = -2$. The term $n$ has been "moved" from the right to the left by combining with $3n$.
- Now, subtract the constant term $10$ from both sides: $2n + 10 - 10 = -2 - 10$, which simplifies to $2n = -12$.
- Finally, isolate $n$ by dividing both sides by the coefficient $2$: $n = -6$.
In formulas, such as the area of a rectangle $A = lw$, $A$, $l$, and $w$ are all terms. Understanding them as terms helps when you rearrange the formula to solve for length: $l = \frac{A}{w}$.
Common Questions About Algebraic Terms
No. A term can be a single number (constant), a single variable, or a combination of both through multiplication. $8$, $x$, and $-5a^2b^3$ are all single terms. What defines a single term is that its parts are multiplied together, not added or subtracted.
A term can have a fractional coefficient. For example, $\frac{2}{3}p$ is a perfectly valid term. The coefficient is $\frac{2}{3}$. When combining like terms with fractions, you add/subtract the fractions as you normally would. Also, variables in the denominator, like $\frac{5}{x}$, are not considered single terms in basic polynomial algebra because they can be rewritten as $5x^{-1}$ (a negative exponent).
Absolutely. Terms can have multiple variables. For instance, $-4xy$, $p^2q^3r$, and $7abc$ are all single terms. They are the product of the coefficient and several variables. Their degree is the sum of the exponents of all these variables.
Footnote
[1] Coefficient (English): A numerical or constant quantity placed before and multiplying the variable in an algebraic term.
[2] Exponent (English): A symbol written above and to the right of a mathematical expression to indicate the power to which it is to be raised.
[3] Degree (of a term) (English): The sum of the exponents of all the variables in a monomial (single term).