MathsCBSE Class 8 · We Distribute, Yet Things Multiply

Distributive Property and Expanding Products for Class 8

Updated 17 October 2026 · Fact-checked

The distributive property says a(b + c) = ab + ac, and applying it twice gives (a + m)(b + n) = ab + an + mb + mn, so each term of one bracket multiplies each term of the other. It explains how a product changes when its numbers change, and it works with negative numbers. Exams ask for expansions with like terms, changes in products, and pattern counting.

Each term of one bracket multiplies each term of the other, and the four small areas add up to the whole product.

Illustration drawn for QuestionBuddy.

Understand Distributive Property and Expanding Products

The distributive property. For any numbers $a, b, c$ : $a (b + c) = ab + a c$ (multiplication distributes over addition), and similarly $(a + b) c = a c + b c$ . Since subtraction is addition of a negative number, $a (b - c) = ab - a c$ too. Integers satisfy the same property, so everything below also holds for negative letter-numbers.

How a product changes. Take the product $23 \times 27$ .

If one factor goes up by 1: $a (b + 1) = ab + a$ , so the product increases by $a$ (here 23).
If both go up by 1: $(a + 1) (b + 1) = ab + a + b + 1$ , so the product increases by $a + b + 1$ .
If one goes up by 1 and the other comes down by 1: $(a + 1) (b - 1) = ab + b - a - 1$ . This can increase or decrease the product, depending on whether $b$ is larger than $a + 1$ or not.

Identity 1. For any $a, b, m, n$ :
$ $(a + m) (b + n) = ab + an + mb + mn$ $
so the change in the product is $an + bm + mn$ . Each term of the first bracket multiplies each term of the second. The identity works when $m$ or $n$ is negative: with $m = 1$ and $n = - 1$ it reproduces $(a + 1) (b - 1)$ . In particular, $(a + u) (b - v) = ab - a v + u b - uv$ , $(a - u) (b + v) = ab + a v - u b - uv$ and $(a - u) (b - v) = ab - a v - u b + uv$ .

An identity is an equation of two algebraic expressions that are equal for all values of their letter-numbers (for integers in this chapter).

Expanding larger brackets. The distributive property is not limited to two terms. To expand $(a + b) (a^{2} + 2 ab + b^{2})$ , multiply each term of $(a + b)$ with each term of the second bracket: $a^{3} + 2 a^{2} b + a b^{2} + a^{2} b + 2 a b^{2} + b^{3}$ . Then add like terms (terms with exactly the same letter-numbers): $a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$ . Terms with different letter-numbers cannot be combined.

Patterns and different expressions. The same pattern can be counted in different ways, giving different-looking expressions; if each is correct they simplify to the same expression. For instance in a pattern whose step $k$ has $k^{2} + 2 k$ circles, the expressions $k (k + 1) + k$ and $(k + 1)^{2} - 1$ both simplify to $k^{2} + 2 k$ .

History. The chapter notes that the distributive law was implicit in the calculations of many ancient civilisations; it mentions Euclid (in geometric form) and Aryabhata (in algebraic form) as using it.

The special identities for squares and the product of a sum and a difference, with fast multiplication, are on the next topic page of this chapter.

Distributive Property and Expanding Products: formulas to remember

Distributive property: $a (b + c) = ab + a c$; Also (a + b)c = ac + bc and a(b - c) = ab - ac.
Identity 1: $(a + m) (b + n) = ab + an + mb + mn$; Valid for positive and negative values of m and n.
Change in a product: $(a + m) (b + n) - ab = an + bm + mn$; The increase when a grows by m and b grows by n.
Both numbers increased by 1: $(a + 1) (b + 1) = ab + a + b + 1$; The product rises by a + b + 1.
One up, one down by 1: $(a + 1) (b - 1) = ab + b - a - 1$; Can increase or decrease the product.

How to solve Distributive Property and Expanding Products questions

Expand and simplify products with this routine.

1Write each bracket as a sum of terms, with the sign attached to each term.
2Multiply every term of the first bracket with every term of the second; the number of products is the product of the numbers of terms.
3Apply the rules of signs for each product, and write letter-numbers in alphabetical order.
4Collect like terms (the same letter-numbers with the same powers).
5Check by substituting a small number such as 1 or 2 for each letter-number in the original and the expanded form.
6For "change in a product" questions, use Identity 1 with m and n signed.

Distributive Property and Expanding Products 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.

One number up, one down

Suppose a shopkeeper plans to sell 23 boxes at Rs 27 each, but instead sells 24 boxes at Rs 26 each.

By how much does the total sale change, and why?

Try it first, then show the solutionSolution

Planned total $= 23 \times 27 = 621$ . New total $= 24 \times 26 = 624$ .
By the identity, $(a + 1) (b - 1) = ab + b - a - 1 = 621 + 27 - 23 - 1 = 624$ .
The change is $b - a - 1 = 27 - 23 - 1 = + 3$ .

Answer: The total rises by Rs 3, to Rs 624.

Memory hook: When the numbers are close, shifting one unit from one factor to the other changes the product only slightly.

Mental multiplication by splitting

Imagine finding the cost of 28 notebooks at Rs 41 each without a calculator.

Use 25 x 38 = 950 and the product-change identity to find 28 x 41.

Try it first, then show the solutionSolution

$28 = 25 + 3$ and $41 = 38 + 3$ , so $m = n = 3$ .
Change $= an + bm + mn = 25 \times 3 + 38 \times 3 + 9 = 75 + 114 + 9 = 198$ .
$28 \times 41 = 950 + 198 = 1148$ .

Answer: Rs 1148

Memory hook: Start from a known product and add the three cross terms.

Tricks to remember Distributive Property and Expanding Products

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

Every term meets every term
In (a + m)(b + n) each term of the first bracket multiplies each term of the second: 4 products from 2 terms and 2 terms.
Use it when: Avoiding missing cross terms.
Signs travel with terms
Write (a - u) as (a + (-u)) before expanding.
Use it when: Expansions with subtraction.
Like terms have the same letters and powers
a squared b and a b squared are different terms and cannot be added.
Use it when: Simplifying after expansion.
Check with 1 and 2
Substitute 1 and 2 for the letters in both forms; equal values give confidence.
Use it when: Verifying expansions in exams.

Distributive Property and Expanding Products: solved examples

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

Example 1: Expanding with signs · 1 mark · MCQBoard level

The expansion of $(a + 3) (b - 2)$ is: (a) $ab - 6$ (b) $ab - 2 a + 3 b - 6$ (c) $ab + 2 a - 3 b - 6$ (d) $ab - 2 a - 3 b + 6$

Try it first, then show the solutionSolution

$(a + 3) (b - 2) = ab - 2 a + 3 b - 6$ .
At $a = 1, b = 1$ : $(4) (- 1) = - 4$ and $1 - 2 + 3 - 6 = - 4$ .

Answer: (b) ab - 2a + 3b - 6

Example 2: Change in a product · 2 marks · Short answerBoard level

The product ab is changed by increasing a by 4 and decreasing b by 3. Find the change in the product in terms of a and b, and its value for a = 23, b = 27.

Try it first, then show the solutionSolution

$(a + 4) (b - 3) = ab - 3 a + 4 b - 12$ , so the change is $- 3 a + 4 b - 12$ . (1 mark)
For $a = 23, b = 27$ : $- 69 + 108 - 12 = 27$ . Check: $27 \times 24 = 648$ and $23 \times 27 = 621$ , difference 27. (1 mark)

Answer: Change = -3a + 4b - 12; for a = 23, b = 27 the product rises by 27.

Example 3: Expanding a binomial and a trinomial · 3 marks · Short answerHarder board level

Expand and simplify $(2 x + 3 y) (x^{2} - x y + y^{2})$ .

Try it first, then show the solutionSolution

Multiply: $2 x \cdot x^{2} + 2 x (- x y) + 2 x \cdot y^{2} + 3 y \cdot x^{2} + 3 y (- x y) + 3 y \cdot y^{2}$ . (1 mark)
$= 2 x^{3} - 2 x^{2} y + 2 x y^{2} + 3 x^{2} y - 3 x y^{2} + 3 y^{3}$ . (1 mark)
Combine like terms: $2 x^{3} + x^{2} y - x y^{2} + 3 y^{3}$ . Check at $x = 2, y = 1$ : $(7) (3) = 21$ and $16 + 4 - 2 + 3 = 21$ . (1 mark)

Answer: 2x^3 + x^2y - xy^2 + 3y^3

Example 4: Using Identity 1 · 3 marks · Short answerHarder board level

Given 25 x 38 = 950, find 28 x 41 by using the change in the product when both numbers are increased by 3, without multiplying 28 by 41 directly.

Try it first, then show the solutionSolution

$28 = 25 + 3$ and $41 = 38 + 3$ , so $m = n = 3$ . (1 mark)
Change $= an + bm + mn = 25 \times 3 + 38 \times 3 + 3 \times 3 = 75 + 114 + 9 = 198$ . (1 mark)
$28 \times 41 = 950 + 198 = 1148$ . (1 mark)

Answer: 1148

Example 5: Different expressions for one pattern · 4 marks · Long answerHarder board level

In a pattern of square tiles, step n is a square frame of tiles around an empty square of side n, so the outer square has side n + 2. Three students write the number of tiles as (a) (n + 2)^2 - n^2 (b) 4(n + 1) (c) 4n + 4. Show that all three are equal, and find the number of tiles in step 10 and step 25.

Try it first, then show the solutionSolution

Expand (a): $(n + 2)^{2} - n^{2} = n^{2} + 4 n + 4 - n^{2} = 4 n + 4$ . (1 mark)
(b) $4 (n + 1) = 4 n + 4$ by the distributive property, and (c) is already $4 n + 4$ . So all three are equal. (1 mark)
Step 10: $4 \times 10 + 4 = 44$ . (1 mark)
Step 25: $4 \times 25 + 4 = 104$ . (1 mark)

Answer: All equal 4n + 4; 44 tiles in step 10 and 104 tiles in step 25.

Where students lose marks in Distributive Property and Expanding Products

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

Mistake: Forgetting the cross terms
The square of a sum is not the sum of the squares; the product has four terms.
Wrong working
" $(a + 3) (b + 2) = ab + 6$ "
Right working
$(a + 3) (b + 2) = ab + 2 a + 3 b + 6$ .
Do this: Count the number of products: two terms times two terms gives four.
Usually costs: 2 marks.
Mistake: Sign error when multiplying a negative term
A negative times a positive is negative; negative times negative is positive.
Wrong working
" $(a - 2) (b - 3) = ab - 3 a - 2 b - 6$ "
Right working
$(a - 2) (b - 3) = ab - 3 a - 2 b + 6$ .
Do this: Write each product with its sign before simplifying.
Usually costs: 1 mark.
Mistake: Combining unlike terms
Only terms with the same letter-numbers and powers can be added.
Wrong working
" $a^{2} b + a b^{2} = 2 a^{2} b^{2}$ "
Right working
$a^{2} b$ and $a b^{2}$ are different terms; the sum stays $a^{2} b + a b^{2}$ .
Do this: Compare letters and powers before adding.
Usually costs: 1 mark.
Mistake: Missing the mn term in the change in product
The increase is an + bm + mn, not just an + bm.
Wrong working
"28 x 41 is 25 x 38 plus 25 x 3 plus 38 x 3 = 950 + 189."
Right working
Add $mn = 9$ also: $950 + 189 + 9 = 1148$ .
Do this: List the three terms an, bm and mn on separate lines.
Usually costs: The final mark.

Exam tips

Write the number of terms in each bracket and the number of products you expect; tick them off as you multiply.
Keep letter-numbers in alphabetical order inside each term so like terms are easy to spot.
Verify each expansion with one small substitution.
For "change in product" answers, give both the algebraic change and the numerical value.
For pattern questions, show that different expressions are equal by simplifying each to the same form.

Practice questions on Distributive Property and Expanding Products

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 · Assertion-reasonBoard level
Assertion (A): (a + 1)(b + 1) = ab + a + b + 1.
Reason (R): It follows by applying the distributive property twice.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Treat (a + 1) as one term first.
Answer
(a). $(a + 1) (b + 1) = (a + 1) b + (a + 1) \cdot 1 = ab + b + a + 1$ . The distributive property is applied twice.
Q2 · 2 marks · Very short answerBoard level
Expand and simplify $(3 p - 2) (p + 5)$ .
Hint
Four products, then combine the p terms.
Answer
$3 p \cdot p + 3 p \cdot 5 - 2 \cdot p - 2 \cdot 5 = 3 p^{2} + 15 p - 2 p - 10 = 3 p^{2} + 13 p - 10$ .
Q3 · 3 marks · Short answerHarder board level
The product 23 x 27 is changed by increasing one number by 5 and decreasing the other by 5. Find the change in terms of a and b, evaluate it for a = 23, b = 27, and state when the product stays unchanged.
Hint
Expand (a + 5)(b - 5).
Answer
$(a + 5) (b - 5) = ab - 5 a + 5 b - 25$ , so the change is $5 (b - a - 5)$ . For $a = 23, b = 27$ : $5 (27 - 23 - 5) = - 5$ , so the product falls from 621 to 616 (check $28 \times 22 = 616$ ). The product is unchanged when $b - a - 5 = 0$ , i.e. $b = a + 5$ , for example $10 \times 15 = 15 \times 10$ .
Q4 · 4 marks · Long answerHOTS / competency
In a calendar, take any 3 x 3 square of dates, with the top-left number a (so the dates are a, a + 1, a + 2, then a + 7, a + 8, a + 9, then a + 14, a + 15, a + 16). Compare the products of the two diagonal corner pairs and explain what you find.
Hint
The corners are a, a + 2, a + 14, a + 16.
Answer
One diagonal gives $a (a + 16) = a^{2} + 16 a$ ; the other gives $(a + 2) (a + 14) = a^{2} + 14 a + 2 a + 28 = a^{2} + 16 a + 28$ . The difference is always 28. Example: $a = 3$ gives $3 \times 19 = 57$ and $5 \times 17 = 85$ , and $85 - 57 = 28$ .
Q5 · 4 marks · Case-basedHOTS / competency
Case study: in a multiplication table with rows and columns numbered 1 to 10, a 3 x 3 frame lies entirely inside the table and has the product pq at its centre. The entry above is (p - 1)q, below is (p + 1)q, left is p(q - 1) and right is p(q + 1).
(i) Write the four corner entries as products. (1 mark)
(ii) Show that the sum of the four corner entries is 4pq. (2 marks)
(iii) The four corners add up to 252. Find the centre entry and the possible p and q. (1 mark)
Hint
The corners are (p - 1)(q - 1), (p - 1)(q + 1), (p + 1)(q - 1), (p + 1)(q + 1).
Answer
(i) The corners are $(p - 1) (q - 1)$ , $(p - 1) (q + 1)$ , $(p + 1) (q - 1)$ and $(p + 1) (q + 1)$ . (ii) Expanding: $pq - p - q + 1$ , $pq + p - q - 1$ , $pq - p + q - 1$ and $pq + p + q + 1$ . Adding, the $p$ , $q$ and constant terms cancel in pairs and the sum is $4 pq$ . (iii) $4 pq = 252$ gives the centre $pq = 63$ , so in a table whose factors go from 1 to 10, with the frame inside the table (p and q between 2 and 9), $(p, q) = (7, 9)$ or $(9, 7)$ .

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.State the distributive property.Show answer
a(b + c) = ab + ac.
2.What is the increase in ab when both a and b increase by 1?Show answer
a + b + 1.
3.Expand (a + m)(b + n).Show answer
ab + an + mb + mn.
4.What are like terms?Show answer
Terms with exactly the same letter-numbers and powers.
5.What is an identity?Show answer
An equation of two algebraic expressions that are equal for all values of the letter-numbers.
6.Does the distributive property hold for negative numbers?Show answer
Yes, for all integers.

Distributive Property and Expanding Products: frequently asked questions

How many terms does (a + b + c)(d + e) have before simplification?

Six products: each of the three terms of the first bracket times each of the two terms of the second.

Is (a + b)(a + b) the same as a squared plus b squared?

No. It is a squared plus 2ab plus b squared; the middle term comes from the cross products ab and ba.

Can the product decrease when one number increases?

Yes, if the other number decreases by more: (a + 1)(b - 1) = ab + b - a - 1 falls when b is less than a + 1.

Why do different pattern expressions look different but agree?

They count the same objects by different groupings; once simplified, they are the same expression.