NCERT Textbook: We Distribute, Yet Things Multiply | Mathematics Class 8- New NCERT (Ganita Prakash) PDF Download

 Page 1


6
WE DISTRIBUTE, 
YET THINGS 
MULTIPLY
We have seen how algebra makes use of letter symbols to write general 
statements about patterns and relations in a compact manner. Algebra 
can also be used to justify or prove claims and conjectures (like the many 
properties you saw in the previous chapter) and to solve problems of 
various kinds. 
Distributivity is a property relating multiplication and addition that 
is captured concisely using algebra. In this chapter, we explore different 
types of multiplication patterns and show how they can be described in 
the language of algebra by making use of distributivity.
6.1 Some Properties of Multiplication
Increments in Products
Consider the multiplication of two numbers, say, 23 × 27.
1.  By how much does the product increase if the first number (23) is 
increased by 1?
2.  What if the second number (27) is increased by 1?
3.  How about when both numbers are increased by 1?
Do you see a pattern that could help generalise our observations to the 
product of any two numbers?
Let us first consider a simpler problem — find the increase in the 
product when 27 is increased by 1. From the definition of multiplication 
(and the commutative property), it is clear that the product increases by 
23. This can be seen from the distributive property of multiplication as 
well. If a, b and c are three numbers, then —
Chapter 6 We Distribute, Yet Things Multiply.indd   136 Chapter 6 We Distribute, Yet Things Multiply.indd   136 10-07-2025   15:10:48 10-07-2025   15:10:48
Page 2


6
WE DISTRIBUTE, 
YET THINGS 
MULTIPLY
We have seen how algebra makes use of letter symbols to write general 
statements about patterns and relations in a compact manner. Algebra 
can also be used to justify or prove claims and conjectures (like the many 
properties you saw in the previous chapter) and to solve problems of 
various kinds. 
Distributivity is a property relating multiplication and addition that 
is captured concisely using algebra. In this chapter, we explore different 
types of multiplication patterns and show how they can be described in 
the language of algebra by making use of distributivity.
6.1 Some Properties of Multiplication
Increments in Products
Consider the multiplication of two numbers, say, 23 × 27.
1.  By how much does the product increase if the first number (23) is 
increased by 1?
2.  What if the second number (27) is increased by 1?
3.  How about when both numbers are increased by 1?
Do you see a pattern that could help generalise our observations to the 
product of any two numbers?
Let us first consider a simpler problem — find the increase in the 
product when 27 is increased by 1. From the definition of multiplication 
(and the commutative property), it is clear that the product increases by 
23. This can be seen from the distributive property of multiplication as 
well. If a, b and c are three numbers, then —
Chapter 6 We Distribute, Yet Things Multiply.indd   136 Chapter 6 We Distribute, Yet Things Multiply.indd   136 10-07-2025   15:10:48 10-07-2025   15:10:48
We Distribute, Yet Things Multiply
137
This is called the distributive property of multiplication over 
addition. Using the identity a (b + c) = ab + ac with a = 23, b = 27, and  
c = 1, we have
Remember that here, a (b + c) and 23 (27 + 1) mean a × (b + c), and  
23 × (27 + 1), respectively. We usually skip writing the ‘×’ symbol before 
or after brackets, just as in the case of expressions like 5a, xy, etc.
We can also similarly expand (a + b) c using the distributive property 
as follows — 
   (a + b) c = c (a + b) (commutativity of multiplication)
       = ca + cb (distributivity)
       = ac + bc (commutativity of multiplication)
We can use the distributive property to find, in general, how much a 
product increases if one or both the numbers in the product are increased 
by 1. Suppose the initial two numbers are a and b. If one of the numbers, 
say b, is increased by 1, then we have —
Now let us see what happens if both numbers in a product are 
increased by 1. If in a product  ab, both a and b are increased by 1, then  
we obtain (a + 1) (b + 1).
a (b + c) = ab + ac                   a (b + c)
           . . . . .  
           . . . . .  
           . . . . . 
.   .   .            .    .
.   .   .            .    .
.   .   .            .    .
.   .   .            .    .
           . . . . .  
           . . . . .
            . . . . . .  
            . . . . . .
            . . . . . . 
    .   .                 .
    .   .                 .
    .   .                 .
    .   .                 .
            . . . . . . 
            . . . . . .
ab ac
b columns c columns
a rows
This property can be visualised nicely using a diagram:
Increase
23 ( 27 + 1) = 23 × 27 + 23
Increase
a ( b + 1) = ab ×  a
Chapter 6 We Distribute, Yet Things Multiply.indd   137 Chapter 6 We Distribute, Yet Things Multiply.indd   137 10-07-2025   15:10:48 10-07-2025   15:10:48
Page 3


6
WE DISTRIBUTE, 
YET THINGS 
MULTIPLY
We have seen how algebra makes use of letter symbols to write general 
statements about patterns and relations in a compact manner. Algebra 
can also be used to justify or prove claims and conjectures (like the many 
properties you saw in the previous chapter) and to solve problems of 
various kinds. 
Distributivity is a property relating multiplication and addition that 
is captured concisely using algebra. In this chapter, we explore different 
types of multiplication patterns and show how they can be described in 
the language of algebra by making use of distributivity.
6.1 Some Properties of Multiplication
Increments in Products
Consider the multiplication of two numbers, say, 23 × 27.
1.  By how much does the product increase if the first number (23) is 
increased by 1?
2.  What if the second number (27) is increased by 1?
3.  How about when both numbers are increased by 1?
Do you see a pattern that could help generalise our observations to the 
product of any two numbers?
Let us first consider a simpler problem — find the increase in the 
product when 27 is increased by 1. From the definition of multiplication 
(and the commutative property), it is clear that the product increases by 
23. This can be seen from the distributive property of multiplication as 
well. If a, b and c are three numbers, then —
Chapter 6 We Distribute, Yet Things Multiply.indd   136 Chapter 6 We Distribute, Yet Things Multiply.indd   136 10-07-2025   15:10:48 10-07-2025   15:10:48
We Distribute, Yet Things Multiply
137
This is called the distributive property of multiplication over 
addition. Using the identity a (b + c) = ab + ac with a = 23, b = 27, and  
c = 1, we have
Remember that here, a (b + c) and 23 (27 + 1) mean a × (b + c), and  
23 × (27 + 1), respectively. We usually skip writing the ‘×’ symbol before 
or after brackets, just as in the case of expressions like 5a, xy, etc.
We can also similarly expand (a + b) c using the distributive property 
as follows — 
   (a + b) c = c (a + b) (commutativity of multiplication)
       = ca + cb (distributivity)
       = ac + bc (commutativity of multiplication)
We can use the distributive property to find, in general, how much a 
product increases if one or both the numbers in the product are increased 
by 1. Suppose the initial two numbers are a and b. If one of the numbers, 
say b, is increased by 1, then we have —
Now let us see what happens if both numbers in a product are 
increased by 1. If in a product  ab, both a and b are increased by 1, then  
we obtain (a + 1) (b + 1).
a (b + c) = ab + ac                   a (b + c)
           . . . . .  
           . . . . .  
           . . . . . 
.   .   .            .    .
.   .   .            .    .
.   .   .            .    .
.   .   .            .    .
           . . . . .  
           . . . . .
            . . . . . .  
            . . . . . .
            . . . . . . 
    .   .                 .
    .   .                 .
    .   .                 .
    .   .                 .
            . . . . . . 
            . . . . . .
ab ac
b columns c columns
a rows
This property can be visualised nicely using a diagram:
Increase
23 ( 27 + 1) = 23 × 27 + 23
Increase
a ( b + 1) = ab ×  a
Chapter 6 We Distribute, Yet Things Multiply.indd   137 Chapter 6 We Distribute, Yet Things Multiply.indd   137 10-07-2025   15:10:48 10-07-2025   15:10:48
Ganita Prakash | Grade 8 
138
How do we expand this?
Let us consider (a + 1) as a single term. Then, by the distributive 
property, we have
Thus, the product ab increases by a + b + 1 when each of a and b are 
increased by 1.
What would we get  if we had expanded (a + 1) (b + 1) by first taking ( b + 1)  
as a single term? Try it?
What happens when one of the numbers in a product is increased by 1 
and the other is decreased by 1? Will there be any change in the product? 
Let us again take the product ab of two numbers a and b. If a is 
increased by 1 and b is decreased by 1, then their product will be (a + 1) 
(b – 1). Expanding this, we get
Will the product always increase? Find 3 examples where the product 
decreases.
What happens when a and b are negative integers? 
Check by substituting different values for a and b in each of the above 
cases. For example, a = –5, b = 8; a = –4, b = –5; etc.
We have seen that integers also satisfy the distributive property, that 
is, if x, y and z are any three integers, then x (y + z) = xy + xz. 
Thus, the expressions we have for increase of products hold when the 
letter-numbers take on negative integer values as well.
Recall that two algebraic expressions are equal if they take on the 
same values when their letter-numbers are replaced by numbers. These 
(a + 1) (b + 1) = (a + 1) b + (a + 1) 1
   = ab +  (b + a + 1)
Increase
(a + 1) (b + 1) = (a + 1) b + (a + 1) 1
   Again applying the distributive 
property, we obtain
(23 + 1) (27 + 1) = (23 + 1) 27 + (23 + 1) 1
        = 23 × 27 +  (27 + 23 + 1)
Increase
If a = 23, and, b = 27, we get
(a + 1) (b – 1) = (a + 1) b – (a + 1) 1
   = ab +  b – (a + 1)
   = ab +  b – a – 1
Increase
(23 + 1) (27 – 1) = (23 + 1) 27 – (23 + 1) 1
       = 23 × 27 + 27 – (23 + 1)
       = 23 × 27 +  27 – 23 – 1
Increase
If a = 23, and b = 27, we get
Chapter 6 We Distribute, Yet Things Multiply.indd   138 Chapter 6 We Distribute, Yet Things Multiply.indd   138 10-07-2025   15:10:49 10-07-2025   15:10:49
Page 4


6
WE DISTRIBUTE, 
YET THINGS 
MULTIPLY
We have seen how algebra makes use of letter symbols to write general 
statements about patterns and relations in a compact manner. Algebra 
can also be used to justify or prove claims and conjectures (like the many 
properties you saw in the previous chapter) and to solve problems of 
various kinds. 
Distributivity is a property relating multiplication and addition that 
is captured concisely using algebra. In this chapter, we explore different 
types of multiplication patterns and show how they can be described in 
the language of algebra by making use of distributivity.
6.1 Some Properties of Multiplication
Increments in Products
Consider the multiplication of two numbers, say, 23 × 27.
1.  By how much does the product increase if the first number (23) is 
increased by 1?
2.  What if the second number (27) is increased by 1?
3.  How about when both numbers are increased by 1?
Do you see a pattern that could help generalise our observations to the 
product of any two numbers?
Let us first consider a simpler problem — find the increase in the 
product when 27 is increased by 1. From the definition of multiplication 
(and the commutative property), it is clear that the product increases by 
23. This can be seen from the distributive property of multiplication as 
well. If a, b and c are three numbers, then —
Chapter 6 We Distribute, Yet Things Multiply.indd   136 Chapter 6 We Distribute, Yet Things Multiply.indd   136 10-07-2025   15:10:48 10-07-2025   15:10:48
We Distribute, Yet Things Multiply
137
This is called the distributive property of multiplication over 
addition. Using the identity a (b + c) = ab + ac with a = 23, b = 27, and  
c = 1, we have
Remember that here, a (b + c) and 23 (27 + 1) mean a × (b + c), and  
23 × (27 + 1), respectively. We usually skip writing the ‘×’ symbol before 
or after brackets, just as in the case of expressions like 5a, xy, etc.
We can also similarly expand (a + b) c using the distributive property 
as follows — 
   (a + b) c = c (a + b) (commutativity of multiplication)
       = ca + cb (distributivity)
       = ac + bc (commutativity of multiplication)
We can use the distributive property to find, in general, how much a 
product increases if one or both the numbers in the product are increased 
by 1. Suppose the initial two numbers are a and b. If one of the numbers, 
say b, is increased by 1, then we have —
Now let us see what happens if both numbers in a product are 
increased by 1. If in a product  ab, both a and b are increased by 1, then  
we obtain (a + 1) (b + 1).
a (b + c) = ab + ac                   a (b + c)
           . . . . .  
           . . . . .  
           . . . . . 
.   .   .            .    .
.   .   .            .    .
.   .   .            .    .
.   .   .            .    .
           . . . . .  
           . . . . .
            . . . . . .  
            . . . . . .
            . . . . . . 
    .   .                 .
    .   .                 .
    .   .                 .
    .   .                 .
            . . . . . . 
            . . . . . .
ab ac
b columns c columns
a rows
This property can be visualised nicely using a diagram:
Increase
23 ( 27 + 1) = 23 × 27 + 23
Increase
a ( b + 1) = ab ×  a
Chapter 6 We Distribute, Yet Things Multiply.indd   137 Chapter 6 We Distribute, Yet Things Multiply.indd   137 10-07-2025   15:10:48 10-07-2025   15:10:48
Ganita Prakash | Grade 8 
138
How do we expand this?
Let us consider (a + 1) as a single term. Then, by the distributive 
property, we have
Thus, the product ab increases by a + b + 1 when each of a and b are 
increased by 1.
What would we get  if we had expanded (a + 1) (b + 1) by first taking ( b + 1)  
as a single term? Try it?
What happens when one of the numbers in a product is increased by 1 
and the other is decreased by 1? Will there be any change in the product? 
Let us again take the product ab of two numbers a and b. If a is 
increased by 1 and b is decreased by 1, then their product will be (a + 1) 
(b – 1). Expanding this, we get
Will the product always increase? Find 3 examples where the product 
decreases.
What happens when a and b are negative integers? 
Check by substituting different values for a and b in each of the above 
cases. For example, a = –5, b = 8; a = –4, b = –5; etc.
We have seen that integers also satisfy the distributive property, that 
is, if x, y and z are any three integers, then x (y + z) = xy + xz. 
Thus, the expressions we have for increase of products hold when the 
letter-numbers take on negative integer values as well.
Recall that two algebraic expressions are equal if they take on the 
same values when their letter-numbers are replaced by numbers. These 
(a + 1) (b + 1) = (a + 1) b + (a + 1) 1
   = ab +  (b + a + 1)
Increase
(a + 1) (b + 1) = (a + 1) b + (a + 1) 1
   Again applying the distributive 
property, we obtain
(23 + 1) (27 + 1) = (23 + 1) 27 + (23 + 1) 1
        = 23 × 27 +  (27 + 23 + 1)
Increase
If a = 23, and, b = 27, we get
(a + 1) (b – 1) = (a + 1) b – (a + 1) 1
   = ab +  b – (a + 1)
   = ab +  b – a – 1
Increase
(23 + 1) (27 – 1) = (23 + 1) 27 – (23 + 1) 1
       = 23 × 27 + 27 – (23 + 1)
       = 23 × 27 +  27 – 23 – 1
Increase
If a = 23, and b = 27, we get
Chapter 6 We Distribute, Yet Things Multiply.indd   138 Chapter 6 We Distribute, Yet Things Multiply.indd   138 10-07-2025   15:10:49 10-07-2025   15:10:49
We Distribute, Yet Things Multiply
139
numbers could be any integers. Mathematical statements that express 
the equality of two algebraic expressions, such as 
a (b + 8) = ab + 8a, 
(a + 1) (b – 1) = ab + b – a – 1, etc., 
are called identities.
By how much will the product of two numbers change if one of the 
numbers is increased by m and the other by n?
If a and b are the initial numbers being multiplied, they become  
a + m and b + n.
(a + m) (b + n) = (a + m)b + (a + m)n
                          = ab + mb + an + mn
The increase is an + bm + mn.
Notice that the product is the sum of the product of each term of  
(a + m) with each term of (b + n).
This identity can be visualised as follows —
(a + m) (b + n) = ab + mb + an + mn     
Identity 1
b columns n columns
a rows
m rows
(a + m) (b + n)
           . . . . .  
           . . . . .  
           . . . . . 
.   .   .            .    .
.   .   .            .    .
.   .   .            .    .
.   .   .            .    .
           . . . . .  
           . . . . .
ab
            . . . . . .  
            . . . . . .
            . . . . . . 
    .   .                 .
    .   .                 .
    .   .                 .
    .   .                 .
            . . . . . . 
            . . . . . .
an
        . . . . . . .
        . . . . . . . 
.    .                .    .
.    .                .    .
.    .                .    .
        . . . . . .  
mb
        . . . . . . .
        . . . . . . .
 .   .                 .
 .   .                 .
 .   .                 .
         . . . . . . .
mn
Chapter 6 We Distribute, Yet Things Multiply.indd   139 Chapter 6 We Distribute, Yet Things Multiply.indd   139 10-07-2025   15:10:49 10-07-2025   15:10:49
Page 5


6
WE DISTRIBUTE, 
YET THINGS 
MULTIPLY
We have seen how algebra makes use of letter symbols to write general 
statements about patterns and relations in a compact manner. Algebra 
can also be used to justify or prove claims and conjectures (like the many 
properties you saw in the previous chapter) and to solve problems of 
various kinds. 
Distributivity is a property relating multiplication and addition that 
is captured concisely using algebra. In this chapter, we explore different 
types of multiplication patterns and show how they can be described in 
the language of algebra by making use of distributivity.
6.1 Some Properties of Multiplication
Increments in Products
Consider the multiplication of two numbers, say, 23 × 27.
1.  By how much does the product increase if the first number (23) is 
increased by 1?
2.  What if the second number (27) is increased by 1?
3.  How about when both numbers are increased by 1?
Do you see a pattern that could help generalise our observations to the 
product of any two numbers?
Let us first consider a simpler problem — find the increase in the 
product when 27 is increased by 1. From the definition of multiplication 
(and the commutative property), it is clear that the product increases by 
23. This can be seen from the distributive property of multiplication as 
well. If a, b and c are three numbers, then —
Chapter 6 We Distribute, Yet Things Multiply.indd   136 Chapter 6 We Distribute, Yet Things Multiply.indd   136 10-07-2025   15:10:48 10-07-2025   15:10:48
We Distribute, Yet Things Multiply
137
This is called the distributive property of multiplication over 
addition. Using the identity a (b + c) = ab + ac with a = 23, b = 27, and  
c = 1, we have
Remember that here, a (b + c) and 23 (27 + 1) mean a × (b + c), and  
23 × (27 + 1), respectively. We usually skip writing the ‘×’ symbol before 
or after brackets, just as in the case of expressions like 5a, xy, etc.
We can also similarly expand (a + b) c using the distributive property 
as follows — 
   (a + b) c = c (a + b) (commutativity of multiplication)
       = ca + cb (distributivity)
       = ac + bc (commutativity of multiplication)
We can use the distributive property to find, in general, how much a 
product increases if one or both the numbers in the product are increased 
by 1. Suppose the initial two numbers are a and b. If one of the numbers, 
say b, is increased by 1, then we have —
Now let us see what happens if both numbers in a product are 
increased by 1. If in a product  ab, both a and b are increased by 1, then  
we obtain (a + 1) (b + 1).
a (b + c) = ab + ac                   a (b + c)
           . . . . .  
           . . . . .  
           . . . . . 
.   .   .            .    .
.   .   .            .    .
.   .   .            .    .
.   .   .            .    .
           . . . . .  
           . . . . .
            . . . . . .  
            . . . . . .
            . . . . . . 
    .   .                 .
    .   .                 .
    .   .                 .
    .   .                 .
            . . . . . . 
            . . . . . .
ab ac
b columns c columns
a rows
This property can be visualised nicely using a diagram:
Increase
23 ( 27 + 1) = 23 × 27 + 23
Increase
a ( b + 1) = ab ×  a
Chapter 6 We Distribute, Yet Things Multiply.indd   137 Chapter 6 We Distribute, Yet Things Multiply.indd   137 10-07-2025   15:10:48 10-07-2025   15:10:48
Ganita Prakash | Grade 8 
138
How do we expand this?
Let us consider (a + 1) as a single term. Then, by the distributive 
property, we have
Thus, the product ab increases by a + b + 1 when each of a and b are 
increased by 1.
What would we get  if we had expanded (a + 1) (b + 1) by first taking ( b + 1)  
as a single term? Try it?
What happens when one of the numbers in a product is increased by 1 
and the other is decreased by 1? Will there be any change in the product? 
Let us again take the product ab of two numbers a and b. If a is 
increased by 1 and b is decreased by 1, then their product will be (a + 1) 
(b – 1). Expanding this, we get
Will the product always increase? Find 3 examples where the product 
decreases.
What happens when a and b are negative integers? 
Check by substituting different values for a and b in each of the above 
cases. For example, a = –5, b = 8; a = –4, b = –5; etc.
We have seen that integers also satisfy the distributive property, that 
is, if x, y and z are any three integers, then x (y + z) = xy + xz. 
Thus, the expressions we have for increase of products hold when the 
letter-numbers take on negative integer values as well.
Recall that two algebraic expressions are equal if they take on the 
same values when their letter-numbers are replaced by numbers. These 
(a + 1) (b + 1) = (a + 1) b + (a + 1) 1
   = ab +  (b + a + 1)
Increase
(a + 1) (b + 1) = (a + 1) b + (a + 1) 1
   Again applying the distributive 
property, we obtain
(23 + 1) (27 + 1) = (23 + 1) 27 + (23 + 1) 1
        = 23 × 27 +  (27 + 23 + 1)
Increase
If a = 23, and, b = 27, we get
(a + 1) (b – 1) = (a + 1) b – (a + 1) 1
   = ab +  b – (a + 1)
   = ab +  b – a – 1
Increase
(23 + 1) (27 – 1) = (23 + 1) 27 – (23 + 1) 1
       = 23 × 27 + 27 – (23 + 1)
       = 23 × 27 +  27 – 23 – 1
Increase
If a = 23, and b = 27, we get
Chapter 6 We Distribute, Yet Things Multiply.indd   138 Chapter 6 We Distribute, Yet Things Multiply.indd   138 10-07-2025   15:10:49 10-07-2025   15:10:49
We Distribute, Yet Things Multiply
139
numbers could be any integers. Mathematical statements that express 
the equality of two algebraic expressions, such as 
a (b + 8) = ab + 8a, 
(a + 1) (b – 1) = ab + b – a – 1, etc., 
are called identities.
By how much will the product of two numbers change if one of the 
numbers is increased by m and the other by n?
If a and b are the initial numbers being multiplied, they become  
a + m and b + n.
(a + m) (b + n) = (a + m)b + (a + m)n
                          = ab + mb + an + mn
The increase is an + bm + mn.
Notice that the product is the sum of the product of each term of  
(a + m) with each term of (b + n).
This identity can be visualised as follows —
(a + m) (b + n) = ab + mb + an + mn     
Identity 1
b columns n columns
a rows
m rows
(a + m) (b + n)
           . . . . .  
           . . . . .  
           . . . . . 
.   .   .            .    .
.   .   .            .    .
.   .   .            .    .
.   .   .            .    .
           . . . . .  
           . . . . .
ab
            . . . . . .  
            . . . . . .
            . . . . . . 
    .   .                 .
    .   .                 .
    .   .                 .
    .   .                 .
            . . . . . . 
            . . . . . .
an
        . . . . . . .
        . . . . . . . 
.    .                .    .
.    .                .    .
.    .                .    .
        . . . . . .  
mb
        . . . . . . .
        . . . . . . .
 .   .                 .
 .   .                 .
 .   .                 .
         . . . . . . .
mn
Chapter 6 We Distribute, Yet Things Multiply.indd   139 Chapter 6 We Distribute, Yet Things Multiply.indd   139 10-07-2025   15:10:49 10-07-2025   15:10:49
Ganita Prakash | Grade 8 
140
This identity can be used to find how products change when the numbers 
being multiplied are increased or decreased by any amount. Can you see 
how this identity can be used when one or both numbers are decreased?
For example, let us reconsider the case when one number is increased 
by 1 and the other decreased by 1. Let us write the product (a + 1) (b – 1) 
as (a + 1) (b + (–1). Taking m = 1 and n = –1 in Identity 1, we have 
ab + (1) × b + a × (–1) + (1) × (–1) = ab + b – a – 1,
which is the same expression that we obtain earlier. 
Use Identity 1 to find how the product changes when 
(i) one number is decreased by 2 and the other increased by 3;
(ii) both numbers are decreased, one by 3 and the other by 4.  
Verify the answers by finding the products without converting the 
subtractions to additions.
Generalising this, we can find the product ( a + u) (b – v) as follows.
(a + u) (b – v) = (a + u) b – (a + u) v
                        = ab + ub – (av + uv)
                        = ab + ub – av – uv.
Check that this is the same as taking m = u and n = –v in Identity 1.
As in Identity 1, the product (a + u) (b – v) is the sum of the product of 
each term of a + u (a and u) with each term of b – v (b and (–v)). Notice 
that the signs of the terms in the products can be determined using the 
usual rules of integer multiplication.
Expand (i) (a – u) (b + v), (ii) (a – u) (b – v).
We get
        (a – u) (b + v) = ab – ub + av – uv, and
(a – u) (b – v) = ab – ub – av + uv.
The distributive property is not restricted to two terms within a bracket.
Example 1: Expand 
3a
2
 (a – b + 
1
5
).
3a
2
 (a – b + 
1
5
) = (
3a
2
 × a) – (
3a
2
 × b) + (
3a
2
 × 
1
5
 ).
The terms can be simplified as follows —
3a
2
 × a = 
3
2
 × (a × a).
See how the rules of integer multiplication allows us to handle 
multiple cases using a single identity!
Chapter 6 We Distribute, Yet Things Multiply.indd   140 Chapter 6 We Distribute, Yet Things Multiply.indd   140 10-07-2025   15:10:49 10-07-2025   15:10:49