THE distributive property of multiplication it is related to a product in which at least one of the factors is a sum. This property is often used in “head” multiplications, as it is possible to decompose one of the factors to perform this operation more easily. Thus, this property can be applied whenever expressions like the following appear:
a·(b + c)
a, b and c are any real numbers.
The distributive property of multiplication is also called “shower” in Elementary and High School. Next, we will see the practical way to apply this property.
→When only one of the factors is an addition
When only one of the factors is an addition, multiply the other factor by each of its terms and add up the results. In other words:
a·(b + c) = a·b + a·c
Examples:
In the multiplication 10·(2 + 4), we will have:
10·(2 + 4) = 10·2 + 10·4 = 20 + 40 = 60
In the 10·25 multiplication, we will have:
10·25 = 10·(20 + 5) = 200 + 50 = 250
In the multiplication 10·(a + 3), we will have:
10·(a + b) = 10·a + 10·b = 10a +10b
→When the two factors are additions
When two factors are additions, you can apply this property directly or separate it into two cases and then add the results. These alternatives can be mathematically written as follows:
direct form: Each term of the first factor must be multiplied by all the terms of the second factor. All results must be added together at the end. Watch:
(a + b)·(c + d) = a·c + a·d + b·c + b·d
separate form: We write the product of the two additions as the sum of two products. We then solve for each portion of this sum in the way already discussed, for when only one of the terms is an addition. Watch:
(a + b)·(c + d) = a·(c + d) + b·(c + d)
(a + b)·(c + d) = a·c + a·d + b·c + b·d
Examples:
1. In multiplication (2 + 4)·(3+6), we will have:
(2 + 4)·(3+6) = 2·3 + 2·6 + 4·3 + 4·6 = 6 + 12 + 12 + 24 = 54
2. In multiplication (2 + 4)·(7 - 2), we will have:
(2 + 4)·(7 – 2) = 2·7 – 2·2 + 4·7 – 4·2 = 14 – 4 + 28 – 8 = 30
→Additions of three or more installments
When there are three or more installments in any of the factors, proceed in the same way as indicated above. Watch:
(a + b)·(c + d + e) = a·c + a·d + a·e + b·c + b·d + b·e
Example:
In multiplication (2 + 3)·(4 + b + 7), we will have:
(2 + 3)·(4 + b + 7) = 2·4 + 2·b + 2·7 + 3·4 + 3·b + 3·7 =
=8 + 2b + 14 + 12 + 3b + 21 = 55 + 5b
→Multiplications with three or more factors
When there are three or more factors, multiply them two by two, that is, apply the distributive property in the first two and use the result of this multiplication as a factor to apply the same property again. Watch:
(a + b)·(c + d)·(e + f) =
(a·c + a·d + b·c + b·d)·(e + f) =
a·c·e + a·d·e + b·c·e + b·d·e + a·c·f + a·d·f + b·c·f + b·d·f
Example:
In multiplication (2 + 3)·(4 + 5)·(1 + 2), we will have:
(2 + 3)·(4 + 5)·(1 + 2) =
(2·4 + 2·5 + 3·4 + 3·5)·(1 + 2) =
2·4·1 + 2·5·1 + 3·4·1 + 3·5·1 + 2·4·2 + 2·5·2 + 3·4·2 + 3·5·2 =
8 + 10 + 12 + 15 + 16 + 20 + 24 + 30 = 135
Of course, it is also possible to do the sums first and then multiply according to the position of the parentheses. However, when expressions involve unknowns (unknown numbers represented by letters), it is mandatory to perform the multiplication first following this property.
By Luiz Paulo Moreira
Graduated in Mathematics
