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ICSE • Class 7 • Mathematics • Ch 9
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Algebraic Expressions

In ICSE Class 7 Mathematics, "Algebraic Expressions" provides an authoritative, algebraic master study guide analyzing the generalized arithmetic of symbols and operations. This comprehensive chapter explores Basic Concepts (Constants, variables/literals, algebraic terms, algebraic expressions), Classification of Polynomials by Term Count (Monomials [1 term], Binomials [2 terms], Trinomials [3 terms], Polynomials / Multinomials), Factors and Coefficients (Numerical coefficient vs Literal coefficient / variable factor), Like Terms vs Unlike Terms (Combining like terms), Degree of an Algebraic Expression (Highest exponent in single-variable polynomials; Sum of exponents of variables in multi-variable terms), Operations on Algebraic Expressions (Addition and Subtraction using Horizontal and Column/Vertical methods with sign reversals; Multiplication: Monomial by Monomial using exponent product rule, Monomial by Polynomial using the Distributive Law, and Binomial by Binomial), Removal of Brackets using Nested Grouping, and Evaluating Algebraic Expressions by Numerical Substitution aligned with the 2026–27 CISCE curriculum.

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How Did an 8th-Century Baghdad Astronomer Replace Secret Numbers with the Letters of the Alphabet to Solve Impossible Equations?

In the early 9th century in the legendary House of Wisdom in Baghdad, Persian mathematician Muhammad ibn Musa al-Khwarizmi looked at merchant disputes, inheritance calculations, and astrological planetary tables. For thousands of years, ancient mathematicians had solved problems using wordy, convoluted prose. Al-Khwarizmi had a revolutionary breakthrough: he realized you could represent ANY unknown number using an abstract symbol—a variable! He named this science Al-Jabr (from which we get the word Algebra), meaning "the reunion of broken parts"! In algebra, the letter $x$ is not just an alphabet letter; it is a chameleon holding an infinite universe of numbers! Why can you add $5x$ and $3x$ to get $8x$, but trying to add $5x$ and $3y$ leaves you stuck with $5x + 3y$? What is the Degree of a Polynomial with three variables ($4x^2 y^3 z$)? How does the Distributive Law expand expressions? Let's master algebraic expressions.

Why This Chapter Matters

Algebra is the foundational language of all higher mathematics, physics equations, computer programming functions, and financial modeling algorithms. Mastering polynomial terms, coefficients, like-term collection, and polynomial expansion is essential for scoring 100% in ICSE school examinations and future STEM success.

Before You Begin (Prerequisites)

  • Operations on integers (sign rules for addition and multiplication) from Chapter 1.
  • Laws of exponents ($a^m \times a^n = a^{m+n}$) from Chapter 4.
  • Order of operations (BODMAS).

What You Will Learn (Core Objectives)

  • Differentiate between constants, variables, terms, and algebraic expressions.
  • Classify expressions into monomials, binomials, trinomials, and polynomials.
  • Identify numerical and literal coefficients and determine the degree of an expression.
  • Distinguish like terms from unlike terms and simplify by combining like terms.
  • Add and subtract polynomials using both Horizontal and Column methods.
  • Multiply monomials by monomials, monomials by polynomials, and binomials by binomials.
  • Evaluate the numerical value of multi-variable algebraic expressions by substitution.

Chapter Roadmap & Progression

1 1. Anatomy of an Algebraic Expressi...
2 2. Classification of Polynomials &...
3 3. Addition, Subtraction & Multipli...
4 4. Evaluation of Expressions by Sub...

Complete Concept Guide (100% Curriculum Coverage)

1. Anatomy of an Algebraic Expression: Constants & Coefficients

Understand
A. Building Blocks of Algebra:
  • Constant: A symbol having a fixed numerical value (e.g., $5, -12, \frac{3}{4}$).
  • Variable (Literal): A symbol representing an unknown or changing quantity, denoted by letters $x, y, z, a, b$.
  • Algebraic Term: A constant, a variable, or a product of constants and variables separated by $+$ or $-$ signs (e.g., in $7x^2 - 4xy + 9$, the terms are $7x^2$, $-4xy$, and $9$).
  • Factors & Coefficients: In the term $-5xy$:
    • $-5$ is the Numerical Coefficient.
    • $xy$ is the Literal Coefficient (literal factor).
    • The coefficient of $x$ is $-5y$; the coefficient of $y$ is $-5x$.
B. Like vs Unlike Terms:
  • Like Terms: Terms having the exact same literal (variable) factors with the exact same exponents (e.g., $7x^2 y$ and $-4x^2 y$). Only like terms can be combined by adding or subtracting their numerical coefficients!
  • Unlike Terms: Terms having different variables or different powers (e.g., $3x^2 y$ and $3xy^2$). They cannot be combined into a single term!

2. Classification of Polynomials & Degree

Classification & Degree
A. Classification by Number of Terms:
  • Monomial: An expression containing exactly one term (e.g., $5x^3$, $-12ab$, $8$).
  • Binomial: An expression containing exactly two unlike terms (e.g., $3x + 5$, $a^2 - b^2$).
  • Trinomial: An expression containing exactly three unlike terms (e.g., $x^2 + 5x + 6$).
  • Polynomial: An algebraic expression with one or more terms whose variable exponents are strictly non-negative integers (no negative powers or fractional powers like $x^{-1}$ or $\sqrt{x}$).
B. Degree of a Polynomial:
  • Single Variable: The highest power of the variable occurring in the polynomial: $$P(x) = 4x^5 - 7x^3 + 2x - 9 \implies \text{Degree} = \mathbf{5}$$
  • Multiple Variables: For a term with multiple variables, the degree of the term is the sum of the exponents of its variables. The degree of the polynomial is the highest term degree: $$7x^2 y^3 z - 4x^4 y^2 + 8 \implies \text{Term 1: } 2 + 3 + 1 = 6; \quad \text{Term 2: } 4 + 2 = 6$$ The degree of the polynomial is $6$.

3. Addition, Subtraction & Multiplication of Polynomials

Operations
A. Addition & Subtraction:
  • Addition: Group like terms together and add their numerical coefficients.
  • Subtraction: To subtract an expression, change the sign of every term of the expression being subtracted and add to the first expression: $$(5x^2 - 3x + 7) - (2x^2 - 5x - 4) = 5x^2 - 3x + 7 - 2x^2 + 5x + 4 = 3x^2 + 2x + 11$$
B. Multiplication of Expressions:
  1. Monomial by Monomial: Multiply coefficients and add powers of like variables ($a^m \cdot a^n = a^{m+n}$): $$(3x^2 y) \times (-4xy^3) = (3 \times -4) \times (x^{2+1}) \times (y^{1+3}) = \mathbf{-12x^3 y^4}$$
  2. Monomial by Polynomial (Distributive Law): $$a(b + c - d) = ab + ac - ad$$ $$3x(2x^2 - 5x + 4) = 6x^3 - 15x^2 + 12x$$
  3. Binomial by Binomial (FOIL / Distributive): $$(a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd$$

4. Evaluation of Expressions by Substitution

Substitution

To find the numerical value of an algebraic expression for given values of the variables:

  1. Substitute the given numerical value in place of each variable symbol. Use parentheses to protect negative signs!
  2. Follow the BODMAS rule strictly: compute powers/exponents first, then multiplications, and finally additions/subtractions.

Example: Find value of $2x^2 - 3xy + y^2$ when $x = 3$ and $y = -2$:

$$= 2(3)^2 - 3(3)(-2) + (-2)^2$$ $$= 2(9) - 3(-6) + 4 = 18 + 18 + 4 = \mathbf{40}$$

Key Formulas, Identities & Theorems

Degree of Multi-Variable Term
$$\deg(c \cdot x^p y^q z^r) = p + q + r$$
Sum of exponents of all variable factors in the term.
Binomial Multiplication Distributive Expansion
(a + b)(c + d) = ac + ad + bc + bd
First, Outer, Inner, Last (FOIL) expansion.

Algebraic Anatomy, Polynomial Degrees & Multiplication Grid

Algebraic Expressions: Terms, Degree & Operations ANATOMY OF A TERM -7x2y • -7: Numerical Coefficient • x2y: Literal Factor (Variable) • Like Terms: Same variables & powers   (e.g., 3x2y and -5x2y ⇒ Can Add!) • Unlike Terms: Cannot combine!   (e.g., 3x2y and 3xy2 ⇒ Unaltered) POLYNOMIAL DEGREE • Classification by Terms:   Monomial (1) • Binomial (2) • Trinomial (3) • Degree (Highest Power): Single Variable: 5x4 - 3x + 2 ⇒ Degree = 4 Multi-Variable: 4x2y3 + 8xy Term 1 sum = 2 + 3 = 5 ⇒ Degree = 5 • Constant non-zero term has degree 0 OPERATIONS & EVALUATION • Subtraction Rule:   REVERSE ALL SIGNS of lower expression! • Distributive Multiplication: a(b + c) = ab + ac (a+b)(c+d) = ac + ad + bc + bd • Evaluation by Substitution: Replace letters with given numbers Use brackets around negatives: (-2)2=4 • Follow BODMAS hierarchy strictly ONLY LIKE TERMS CAN BE ADDED OR SUBTRACTED • EXPONENTS ADD IN MULTIPLICATION (xm × xn = xm+n)

Chapter Summary & 10 Key Takeaways

Takeaway 1
An algebraic expression is formed by combining constants and variables using addition, subtraction, multiplication, and division.
Takeaway 2
A term is a product of factors; in $-7xy$, $-7$ is the numerical coefficient and $xy$ is the literal coefficient.
Takeaway 3
Like terms possess identical variable factors with identical exponents; only like terms can be added or subtracted.
Takeaway 4
Polynomials are classified by term count: Monomials (1), Binomials (2), Trinomials (3).
Takeaway 5
Degree of a single-variable polynomial is the highest power of the variable.
Takeaway 6
Degree of a multi-variable term is the sum of the powers of all variables in that term.
Takeaway 7
When subtracting polynomials, reverse the sign of every term in the expression being subtracted.
Takeaway 8
When multiplying monomials, multiply numerical coefficients and add powers of matching variables.
Takeaway 9
Multiplication of polynomials relies on the Distributive Law: $a(b + c) = ab + ac$.
Takeaway 10
To evaluate an expression, substitute numerical values inside protective parentheses and follow BODMAS.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
Add the algebraic expressions: $3x^2 - 4xy + 5y^2$, $2y^2 - x^2 + 6xy$, and $5xy - 2y^2 - 2x^2$.
Reveal Answer & Explanation
Answer: Step 1: Group like terms together:
• $x^2$ terms: $3x^2 - x^2 - 2x^2 = (3 - 1 - 2)x^2 = 0x^2 = 0$.
• $xy$ terms: $-4xy + 6xy + 5xy = (-4 + 6 + 5)xy = 7xy$.
• $y^2$ terms: $5y^2 + 2y^2 - 2y^2 = (5 + 2 - 2)y^2 = 5y^2$.
Step 2: Combine the non-zero terms:
$$\text{Sum} = \mathbf{7xy + 5y^2}$$.
Group $x^2$, $xy$, and $y^2$ terms separately. Notice the $x^2$ terms cancel out to 0.
2
Subtract $(2x^3 - 4x^2 + 3x - 5)$ from $(5x^3 - 2x^2 - 8x + 7)$.
Reveal Answer & Explanation
Answer: Step 1: Set up the subtraction and reverse all signs of the expression being subtracted:
$$(5x^3 - 2x^2 - 8x + 7) - (2x^3 - 4x^2 + 3x - 5)$$
$$= 5x^3 - 2x^2 - 8x + 7 - 2x^3 + 4x^2 - 3x + 5$$
Step 2: Collect like terms:
• $x^3$: $5x^3 - 2x^3 = 3x^3$
• $x^2$: $-2x^2 + 4x^2 = +2x^2$
• $x$: $-8x - 3x = -11x$
• Constants: $7 + 5 = +12$
Result: $\mathbf{3x^3 + 2x^2 - 11x + 12}$.
Remember to flip every sign in $(2x^3 - 4x^2 + 3x - 5)$ to $(-2x^3 + 4x^2 - 3x + 5)$, then combine.
3
Multiply the binomials: $(3x - 5y)(2x + 7y)$.
Reveal Answer & Explanation
Answer: Apply the distributive expansion $(a + b)(c + d) = a(c + d) + b(c + d)$:
$$(3x - 5y)(2x + 7y) = 3x(2x + 7y) - 5y(2x + 7y)$$
$$= (3x \times 2x) + (3x \times 7y) - (5y \times 2x) - (5y \times 7y)$$
$$= 6x^2 + 21xy - 10xy - 35y^2$$
Combine middle like terms ($+21xy - 10xy = +11xy$):
$$= \mathbf{6x^2 + 11xy - 35y^2}$$.
Multiply each term in the first binomial by each term in the second: $6x^2 + 21xy - 10xy - 35y^2$.
4
Find the degree of the polynomial: $5x^3 y^2 - 8x^4 y^3 z + 2x^2 y^5 - 11$.
Reveal Answer & Explanation
Answer:

Find the degree of each individual term by summing the powers of all variables in that term:
• Term 1 ($5x^3 y^2$): $3 + 2 = 5$
• Term 2 ($-8x^4 y^3 z^1$): $4 + 3 + 1 = 8$
• Term 3 ($2x^2 y^5$): $2 + 5 = 7$
• Term 4 ($-11$): $0$ (constant term)
The degree of the polynomial is the highest term degree, which is $8$.


Sum variable exponents for each term: $3+2=5$, $4+3+1=8$, $2+5=7$. The highest is 8.
5
If $a = 2$, $b = -3$, and $c = 1$, evaluate the numerical value of: $3a^2 b - 2ab^2 + 4abc - c^3$.
Reveal Answer & Explanation
Answer: Substitute $a = 2$, $b = -3$, and $c = 1$ inside parentheses:
$$3(2)^2(-3) - 2(2)(-3)^2 + 4(2)(-3)(1) - (1)^3$$
Step 1: Compute powers:
$(2)^2 = 4, \quad (-3)^2 = 9, \quad (1)^3 = 1$
Step 2: Compute multiplications:
$$3(4)(-3) = 3 \times (-12) = -36$$
$$-2(2)(9) = -4 \times 9 = -36$$
$$4(2)(-3)(1) = 8 \times (-3) = -24$$
$$-(1)^3 = -1$$
Step 3: Combine all terms:
$$-36 - 36 - 24 - 1 = -(36 + 36 + 24 + 1) = \mathbf{-97}$$.
Square $a$ ($4$) and $b$ ($9$) first. Watch negative signs: $-36 - 36 - 24 - 1 = -97$.
6
Simplify the expression by removing brackets: $4a - [2b - \{3a - (2b - 3a)\}]$.
Reveal Answer & Explanation
Answer: Step 1: Remove round brackets $(2b - 3a)$:
$$-\{3a - 2b + 3a\} = -\{6a - 2b\}$$
Step 2: Remove curly braces:
$$[2b - 6a + 2b] = [4b - 6a]$$
Step 3: Remove square brackets and distribute the negative sign:
$$4a - (4b - 6a) = 4a - 4b + 6a$$
Step 4: Combine like terms:
$$(4a + 6a) - 4b = \mathbf{10a - 4b}$$.
Resolve round brackets: $3a - 2b + 3a = 6a - 2b$. Then resolve curly braces, and finally square brackets.
7
What must be subtracted from $3a^2 - 6ab - 3b^2 - 1$ to get $4a^2 - 7ab - 4b^2 + 1$?
Reveal Answer & Explanation
Answer: Let the required expression to be subtracted be $E$.
$$(3a^2 - 6ab - 3b^2 - 1) - E = (4a^2 - 7ab - 4b^2 + 1)$$
$$E = (3a^2 - 6ab - 3b^2 - 1) - (4a^2 - 7ab - 4b^2 + 1)$$
Change signs of the second expression and combine like terms:
$$E = 3a^2 - 6ab - 3b^2 - 1 - 4a^2 + 7ab + 4b^2 - 1$$
• $a^2$: $3a^2 - 4a^2 = -a^2$
• $ab$: $-6ab + 7ab = +ab$
• $b^2$: $-3b^2 + 4b^2 = +b^2$
• Constants: $-1 - 1 = -2$
Result: $\mathbf{-a^2 + ab + b^2 - 2}$.
Subtract the target from the initial expression: $(3a^2 - 6ab - 3b^2 - 1) - (4a^2 - 7ab - 4b^2 + 1)$.
8
Why is $3x^2 + \frac{5}{x} - 7$ NOT a polynomial?
Reveal Answer & Explanation
Answer:

• Definition of a Polynomial: A polynomial is an algebraic expression in which the exponents of all variable terms must be non-negative integers (i.e., whole numbers $\{0, 1, 2, 3, \dots\}$).
• In the expression $3x^2 + \frac{5}{x} - 7$, the middle term $\frac{5}{x}$ can be written in exponential form as $5x^{-1}$.
• Since the exponent is $-1$ (a negative integer), it violates the non-negative integer condition. Therefore, it is an algebraic expression, but NOT a polynomial.


The term $5/x = 5x^{-1}$ contains a negative exponent ($-1$); polynomials only allow non-negative integer powers.
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