The distributive property, which is also known as the distributive law of multiplication over addition and subtraction, is a property that can be found throughout mathematics. Simply by looking at the name, it is clear that the action involves dividing or distributing something. Additionally, the distributive principle of addition over multiplication is referred to as this formula. Let's look at several examples of the distributive property that have been solved.
Any statement containing three numbers A, B, and C, given in the form A (B + C), is resolved as A (B + C) = AB + AC, or A (B – C) = AB – AC, according to the distributive property of arithmetic. Essentially, this means that operand A is shared across the other two operands. Distributivity of multiplication is a feature that is also known as the distributivity of addition or subtraction.
The Distributive Property of Multiplication Over Addition
We apply the distributive property of multiplication over addition when we are obliged to multiply a particular integer by a sum of two numbers in this situation. Consider the following example: 7(20 + 3). According to mathematics, this is represented as 7(20+3). Following the laws of order of operations, we first solve the sum contained between the parenthesis, and then multiply the result by the number seven.
7(20 + 3) = 7(23) = 161 is the sum of 7(20 + 3). If we use the distributive property to solve the statement, we can first multiply every addend by 7. This is referred to as spreading the number 7 among the two addends, and after that, we can proceed to adding the items.
The addition of 7(20) and 7(3) will be preceded by the multiplication of 7(20) and 7(3).
7(20) + 7(3) = 140 + 21 = 161
We can see that the acquired outcome in both situations is the same, whether the experiment is done before or after.
The Distributive Property of Multiplication Over Subtraction
The distributive property of multiplication over addition was discussed in detail in the preceding section. Subtraction will be discussed in detail in this section. There will be no difference in the process other than the presence of a sign.
Now, let's look at an illustration of the distributive property of multiplication over subtraction in practise. Consider the situation where we must multiply 7 by a difference of 20 and 3, i.e. 7(20 – 3).
Let's have a look at two distinct techniques to resolving the problem.
Calculation method 1: 7 x (20-3), which is 7 x 17 = 119
Calculation method 2: 7 x (20– 3) = (7 x 20) – (7 x 3) = 140 – 21 = 119
Polynomial Operations
There is no difference between the two procedures in terms of the ultimate output.
Verification of Distributive Property
Let's try to explain how distributive property works for different operations and provide some justification. Using the three fundamental operations of addition, subtraction, and division, we shall apply the distributive property law to each operation independently.
Distributive Property of Addition: For addition, the distributive property law is represented as A x (B + C) = AB + AC. Let's try to fix a few numbers in the property to see whether the results are the same. As an illustration,
⇒ 10(5 + 9) = 10×5 + 10× 9
⇒ 140 = 140.
LHS = RHS.
Distributive Property of Subtraction: The general distributive property law for subtraction is stated as A (B - C) = AB - AC. Let's try to fix a few numbers in the property to see whether the results are the same. As an illustration,
⇒ 10(9 - 5) = 10×9 - 10× 5
⇒ 40 = 40.
LHS = RHS.
Distributive Property of Division: We may demonstrate the division of bigger numbers using the distributive property by simply dividing the larger number into two or more smaller parts, as shown in the following example. Consider the following scenario as an illustration of the same. 30 divided by 5 equals 6. Divide 30 by 5.
30 can also be written as 20 + 10
30 ÷ 5 = (20 + 10) ÷ 5
Now let us distribute the division operation for each factor (20 and 10) in the bracket;
⇒ 30 ÷ 5 = (20÷5) + (10÷5)
⇒ 6 = 4 + 2
Therefore, 6 = 6
LHS = RHS
Distributive Property for Polynomials
We can apply the distributive property for polynomials also. This can be used to add or even subtract over multiplication. For example:
3x(4x²-5x²) = 3x.4x² - 3x.5x² = 12x3 – 15x³ = -3x³
Moreover, we can even use the distributive law for polynomials in factor form. This can be illustrated as:
(3x – 4x)(6x² - 7x²) = 3x(6x² - 7x²) -4x(6x² - 7x²) = 18x³ - 21x³ -24x³ + 28x³ = x³
Conclusion
The distributive property is a fundamental principle of mathematics that is employed for adding, subtracting, multiplying, and dividing huge amounts of information. By grouping the numbers together, we may break them down into smaller pieces that can be used to answer the larger equations in any sequence. It makes calculations simpler and more efficient.
The distributive law is very useful and it eases the process of multiplication as well as division over addition and subtraction. Also, we can see it frequently in polynomial and algebraic equations to ease the addition and subtraction processes.