Algebraic Expressions • Topic 1 of 3

Terms, Factors & Types of Expressions

What are Algebraic Expressions?

An algebraic expression is a combination of constants, variables, and mathematical operations (addition, subtraction, multiplication, division, exponents). For example: $3x + 5$, $2a^2 - 4ab + b^2$, $7y - 3$

What are Terms, Factors, and Coefficients?

Term: A single number, variable, or product of numbers and variables separated by $+$ or $-$ signs
Example: In $4x^2 + 3xy - 7$, the terms are $4x^2$, $3xy$, and $-7$

Factor: Numbers or variables multiplied together to form a term
Example: In $6xy$, factors are $6$, $x$, and $y$

Coefficient: The numerical factor of a term (the number multiplied by the variable)
Example: In $5x^2$, the coefficient is $5$

Types of Algebraic Expressions:

Type	Definition	Number of Terms	Examples
Monomial	Expression with exactly one term	1	$7x$, $-3y^2$, $5ab$, $12$
Binomial	Expression with exactly two terms	2	$x + y$, $3a^2 - 2b$, $4x^2 + 7$
Trinomial	Expression with exactly three terms	3	$x^2 + 2x + 1$, $a^2 - 2ab + b^2$
Polynomial	Expression with one or more terms	1 or more	All of the above are polynomials

Like Terms vs Unlike Terms:

Like terms: Terms with the same variables raised to the same powers
Example: $3x^2$ and $5x^2$ are like terms; $2xy$ and $7xy$ are like terms

Unlike terms: Terms with different variables or different exponents
Example: $4x^2$ and $4x^3$ are unlike terms; $3x$ and $3y$ are unlike terms

Real-life Example: Think of a shopping bill: $5x$ (5 apples at $x$ rupees each) $+ 3y$ (3 bananas at $y$ rupees each) $+ 10$ (fixed packaging charge). Here $5x$, $3y$, and $10$ are terms; $5$ and $3$ are coefficients.

Solved Examples

Worked Example

Example 1: Identify the terms, coefficients, and constant term in the expression $7x^2 - 3xy + 5y - 9$

Solution- Step 1: Terms are: $7x^2$, $-3xy$, $5y$, and $-9$ - Step 2: Coefficients: For $7x^2$ → $7$; For $-3xy$ → $-3$; For $5y$ → $5$ - Step 3: Constant term (term without variable): $-9$

Worked Example

Example 2: Classify the following expressions as monomial, binomial, trinomial, or polynomial: (i) $5x^3$, (ii) $a^2 + 2ab + b^2$, (iii) $3m - 4n$, (iv) $x^3 + x^2 + x + 1$

Solution- Step 1: $5x^3$ has 1 term → **Monomial - Step 2: $a^2 + 2ab + b^2$ has 3 terms → **Trinomial** (also a polynomial) - Step 3: $3m - 4n$ has 2 terms → **Binomial - Step 4: $x^3 + x^2 + x + 1$ has 4 terms → **Polynomial

Worked Example

Example 3: From the expression $4x^2y - 3xy^2 + 2x^2y - 5y + 7xy^2$, identify like terms and combine them.

Solution- Step 1: List all terms: $4x^2y$, $-3xy^2$, $2x^2y$, $-5y$, $7xy^2$ - Step 2: Identify like terms: - Terms with $x^2y$: $4x^2y$ and $2x^2y$ - Terms with $xy^2$: $-3xy^2$ and $7xy^2$ - Term with $y$ only: $-5y$ (no like term) - Step 3: Combine $x^2y$ terms: $4 + 2 = 6$ → $6x^2y$ - Step 4: Combine $xy^2$ terms: $-3 + 7 = 4$ → $4xy^2$ - Step 5: Keep $-5y$ as is - Step 6: Write simplified expression: $6x^2y + 4xy^2 - 5y$

Key Points

A term is a single part of an expression separated by $+$ or $-$ signs
A coefficient is the numerical factor multiplied by the variable(s) in a term
Monomial: 1 term | Binomial: 2 terms | Trinomial: 3 terms | Polynomial: 1 or more terms
Like terms have identical variable parts (same variables raised to same exponents)
Unlike terms cannot be combined through addition or subtraction
Constants are terms without any variables
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Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.An expression with exactly one term is a:

Explanation: One term.

Q2.An expression with two terms is a:

Explanation: Two terms.

Q3.In $7x$, the coefficient is:

Explanation: $7$.

Q4.Like terms have the same variables with the same:

Explanation: Same powers.

Practice more MCQs → 📝 Assignment · 35 marks

Type	Definition	Number of Terms	Examples
Monomial	Expression with exactly one term	1	\(7x\), \(-3y^2\), \(5ab\), \(12\)
Binomial	Expression with exactly two terms	2	\(x + y\), \(3a^2 - 2b\), \(4x^2 + 7\)
Trinomial	Expression with exactly three terms	3	\(x^2 + 2x + 1\), \(a^2 - 2ab + b^2\)
Polynomial	Expression with one or more terms	1 or more	All of the above are polynomials