In this publication, we will consider one of the abbreviated multiplication formulas, which allows us to decompose the cube of the sum into factors, and also, we will analyze in detail an example of solving the problem.
sum cube formula
Cube of the sum of terms a и b equals a cube a plus triple the product of the square a on b plus triple the product of the square b on a plus cube b.
(a+b)3 = A3 + 3a2b+3ab2 + b3
The formula is equivalent in reverse order:
a3 + 3a2b+3ab2 + b3 = (a + b)3
Proof of the formula
The cube of a number/expression is its raising to the third power. Let’s represent our expression as a cube:
(a+b)3 = (a + b)(a + b)(a + b).
We multiply the brackets, taking into account the arithmetic rules:
(a + b)(a + b)(a + b) = (a + b)(a + b)2 = (a + b)(a2 + 2ab + b2) = the3 + 2a2b+ab2 + a2b+2ab2 + b3 = A3 + 3a2b+3ab2 + b3.
Note: when opening the brackets, the formula of the square of the sum was used:
(a+b)2 = A2 + 2ab + b2.
Example
What is the cube of the sum (5x + 7y)3?
Solution
We use the abbreviated multiplication formula:
(5x + 7y)3 = (5x)3 + 3 (5x)2 ⋅ 7y + 3 ⋅ 5x ⋅ (7y)2 + (7y)3 = 125x3 + 525x2and + 735xy2 + 343y3
inspection
Let’s multiply three identical brackets:
(5x + 7y)3 = (5x + 7y) (5x + 7y) (5x + 7y) = (5x + 7y) (5x + 7y)2 = (5x + 7y)(25x2 + 70xy + 49y2) = 125x3 + 350x2and + 245xy2 + 175x2and + 490xy2 + 343y3 = 125x3 + 525x2and + 735xy2 + 343y3