MathsICSE Class 8 · Algebraic Expressions and Identities

Multiplication and Division of Algebraic Expressions (ICSE Class 8 Maths)

Updated 17 October 2026 · Fact-checked

To multiply, multiply every term of one expression by every term of the other (using $x^m \times x^n = x^{m+n}$) and collect like terms. To divide a polynomial by a monomial, divide each term separately; to divide by a binomial, use long division or factorise. Check by substituting a number.

Understand Multiplication and Division of Algebraic Expressions

The CISCE curriculum covers multiplication and division of algebraic expressions with integer coefficients.

Monomial times monomial. Multiply the coefficients and add the powers of the same variable: $(3 x^{2}) (- 4 x^{3}) = - 12 x^{5}$ .

Polynomial times monomial. Distribute: $4 x (3 x^{2} - 2 x + 5) = 12 x^{3} - 8 x^{2} + 20 x$ .

Polynomial times polynomial. Multiply each term of the first by each term of the second, then combine like terms: $(2 x - 3) (x^{2} + 4 x - 5) = 2 x^{3} + 8 x^{2} - 10 x - 3 x^{2} - 12 x + 15 = 2 x^{3} + 5 x^{2} - 22 x + 15$ .

Division by a monomial. Divide each term of the numerator by the monomial and use $x^{m} \div x^{n} = x^{m - n}$ : $\frac{12 x ^{4} y ^{3} - 18 x ^{3} y ^{4} + 6 x ^{2} y ^{2}}{6 x ^{2} y ^{2}} = 2 x^{2} y - 3 x y^{2} + 1$ .

Division by a binomial. Either factorise the numerator and cancel the common factor, or use long division (divide the leading term, multiply back, subtract, repeat). When the division is exact, the quotient times the divisor reproduces the dividend.

Checking. Substitute a simple value (like $x = 1$ or $2$ ) into both sides. It will not prove the identity but exposes most slips.

Sign care. A minus sign in front of a bracket changes the sign of every term inside when removing it.

Multiplication and Division of Algebraic Expressions: formulas to remember

Law of indices for products: $x^{m} \times x^{n} = x^{m + n}$; Add exponents when multiplying.
Law of indices for quotients: $x^{m} \div x^{n} = x^{m - n}$; Subtract exponents when dividing.
Distributive law: $a (b + c) = ab + a c$; Basis of all multiplication of expressions.
Exact division check: $dividend = divisor \times quotient$; Use to verify.

How to solve Multiplication and Division of Algebraic Expressions questions

Multiply term by term, collect like terms, and verify by substitution.

1Multiply the coefficients, then the variables (add exponents).
2For two brackets, multiply each term of one by each term of the other.
3Collect like terms and write in descending powers.
4For division, divide term by term or use long division.
5Check with a value such as $x = 1$ or $x = 2$ .

Multiplication and Division of Algebraic Expressions in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Area of an extended garden

Suppose a rectangular garden is $(3 x - 4)$ m long and $(2 x + 3)$ m wide, where the value of $x$ is such that both are positive.

Write the area as a polynomial and find it for $x = 5$ .

Try it first, then show the solutionSolution

Area $= (3 x - 4) (2 x + 3) = 6 x^{2} + 9 x - 8 x - 12 = 6 x^{2} + x - 12$ .
For $x = 5$ : $6 \times 25 + 5 - 12 = 143$ m $^{2}$ (also $11 \times 13 = 143$ ).

Answer: $6 x^{2} + x - 12$ ; $143$ m $^{2}$ .

Memory hook: Every term meets every term.

Tricks to remember Multiplication and Division of Algebraic Expressions

Short memory aids. Say each one aloud once, then use it on a question the same day.

Every term meets every term
A two-term by three-term product has 6 small products; count them to be sure none is missed.
Use it when: Multiplying polynomials.
Test with x = 1
The sum of coefficients of the product must equal the product of the sums of coefficients.
Use it when: Quick check.

Multiplication and Division of Algebraic Expressions: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Monomial product · 1 mark · MCQBoard level

$(3 x^{2}) (- 4 x^{3})$ equals:
(a) $- 12 x^{6}$ (b) $- 12 x^{5}$ (c) $12 x^{5}$ (d) $- x^{5}$

Try it first, then show the solutionSolution

Coefficients: $3 \times (- 4) = - 12$ . Powers: $2 + 3 = 5$ .

Answer: (b)

Example 2: Binomial times trinomial · 3 marks · Short answerBoard level

Multiply $(2 x - 3)$ by $(x^{2} + 4 x - 5)$ and check at $x = 2$ .

Try it first, then show the solutionSolution

$2 x (x^{2} + 4 x - 5) = 2 x^{3} + 8 x^{2} - 10 x$ .
$- 3 (x^{2} + 4 x - 5) = - 3 x^{2} - 12 x + 15$ .
Sum: $2 x^{3} + 5 x^{2} - 22 x + 15$ .
Check $x = 2$ : $(1) (7) = 7$ and $16 + 20 - 44 + 15 = 7$ .

Answer: $2 x^{3} + 5 x^{2} - 22 x + 15$

Example 3: Simplifying a difference of products · 3 marks · Short answerHarder board level

Simplify $(x + 3) (x - 2) - (x - 4) (x + 1)$ .

Try it first, then show the solutionSolution

$(x + 3) (x - 2) = x^{2} + x - 6$ .
$(x - 4) (x + 1) = x^{2} - 3 x - 4$ .
Difference: $x^{2} + x - 6 - x^{2} + 3 x + 4 = 4 x - 2$ .

Answer: $4 x - 2$

Example 4: Two-variable product · 4 marks · Short answerHarder board level

Multiply $(3 a - 2 b)$ by $(a^{2} + ab - 2 b^{2})$ and verify for $a = 2, b = 1$ .

Try it first, then show the solutionSolution

$3 a \times (a^{2} + ab - 2 b^{2}) = 3 a^{3} + 3 a^{2} b - 6 a b^{2}$ .
$- 2 b \times (a^{2} + ab - 2 b^{2}) = - 2 a^{2} b - 2 a b^{2} + 4 b^{3}$ .
Sum: $3 a^{3} + a^{2} b - 8 a b^{2} + 4 b^{3}$ .
Check: LHS $= (6 - 2) (4 + 2 - 2) = 16$ ; RHS $= 24 + 4 - 16 + 4 = 16$ .

Answer: $3 a^{3} + a^{2} b - 8 a b^{2} + 4 b^{3}$

Example 5: Dividing by a binomial · 4 marks · Case-basedHOTS / competency

The area of a rectangle is $(6 x^{2} + x - 12)$ and its length is $(3 x - 4)$ . Find the width and the perimeter, and verify for $x = 5$ .

Try it first, then show the solutionSolution

Divide: $6 x^{2} \div 3 x = 2 x$ ; $2 x (3 x - 4) = 6 x^{2} - 8 x$ ; subtract: $9 x - 12$ .
$9 x \div 3 x = 3$ ; $3 (3 x - 4) = 9 x - 12$ ; remainder $0$ . Width $= 2 x + 3$ .
Perimeter $= 2 [(3 x - 4) + (2 x + 3)] = 2 (5 x - 1) = 10 x - 2$ .
For $x = 5$ : length $11$ , width $13$ , area $143 = 6 \times 25 + 5 - 12$ .

Answer: Width $2 x + 3$ ; perimeter $10 x - 2$ .

Where students lose marks in Multiplication and Division of Algebraic Expressions

The slips examiners see most often, and what to write instead.

Mistake: Missing a term when expanding brackets
Each term of the first bracket must meet every term of the second.
Wrong working
$(2 x - 3) (x^{2} + 4 x - 5) = 2 x^{3} + 8 x^{2} - 15$ .
Right working
$2 x^{3} + 5 x^{2} - 22 x + 15$ .
Do this: Count the small products before collecting.
Usually costs: Most of the marks.
Mistake: Sign error when removing a bracket preceded by minus
The minus sign changes every term inside.
Wrong working
$x^{2} + x - 6 - (x^{2} - 3 x - 4) = x^{2} + x - 6 - x^{2} - 3 x - 4$ .
Right working
$= x^{2} + x - 6 - x^{2} + 3 x + 4 = 4 x - 2$ .
Do this: Put the bracket back and change all signs.
Usually costs: Answer mark.

Exam tips

Write the middle step with all small products; do not combine in your head.
Check by substitution of a small number.
Present the answer in descending powers.

Practice questions on Multiplication and Division of Algebraic Expressions

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
$(- 5 a^{2} b) (2 a b^{3})$ equals:
(a) $- 10 a^{2} b^{3}$ (b) $- 3 a^{3} b^{4}$ (c) $- 10 a^{3} b^{4}$ (d) $10 a^{3} b^{4}$
Hint
Multiply coefficients, add powers of each letter.
Answer
(c)
Q2 · 2 marks · Short answerBoard level
Multiply $4 x (3 x^{2} - 2 x + 5)$ .
Hint
Distribute 4x to each term.
Answer
$12 x^{3} - 8 x^{2} + 20 x$ .
Q3 · 3 marks · Short answerHarder board level
Multiply $(x^{2} - 2 x + 3)$ by $(x - 1)$ and check at $x = 2$ .
Hint
Six small products.
Answer
$x^{3} - x^{2} - 2 x^{2} + 2 x + 3 x - 3 = x^{3} - 3 x^{2} + 5 x - 3$ . At $x = 2$ : $(3) (1) = 3$ and $8 - 12 + 10 - 3 = 3$ .
Q4 · 3 marks · Short answerHarder board level
Divide $15 p^{3} q^{2} - 10 p^{2} q^{3} + 5 pq$ by $5 pq$ .
Hint
Divide each term by 5pq.
Answer
$3 p^{2} q - 2 p q^{2} + 1$ .
Q5 · 4 marks · Long answerHOTS / competency
Find the expression which, multiplied by $(2 x - 3 y)$ , gives $4 x^{2} - 9 y^{2}$ , and check for $x = 3, y = 1$ .
Hint
Divide, or recognise the difference of squares.
Answer
$\frac{4 x ^{2} - 9 y ^{2}}{2 x - 3 y} = 2 x + 3 y$ . Check: $(6 - 3) (6 + 3) = 27$ and $36 - 9 = 27$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.x^m times x^n?Show answer
x^(m+n).
2.How do you divide a polynomial by a monomial?Show answer
Divide each term separately.
3.(x+3)(x-2)?Show answer
x^2 + x - 6.
4.What can you use to check a product?Show answer
Substitute a number into both sides.

Multiplication and Division of Algebraic Expressions: frequently asked questions

Does a check at one value prove the product?

No, but a mismatch proves an error.

Is division by a binomial always exact?

Not always; Class 8 questions are chosen so that it is.