Resolving techniques of notable products are of great importance in solving expressions where the exponent has a numerical value equal to 3. The expressions (a + b) ³ and (a – b) ³ can be solved by the method of distribution or by the method of practical resolution. We will demonstrate both situations, leaving it up to the student to choose the best way to solve them.
Sum Cube
We have that the expression (a + b) ³ can be written as follows: (a + b) ² * (a + b). Decomposition allows us to apply the square of the sum to the expression (a + b) ², multiplying the result by the expression (a + b). Look:
(a + b) ² = a² + 2ab + b² → (a² + 2ab + b²) * (a + b) = a²*a + a²*b + 2ab*a + 2ab*b + b²*a + b²*b
a³ + a²b + 2a²b + 2ab² + ab² + b³ → a³ + 3a²b + 3ab² + b³
(2x + 3)³ = (2x + 3)² * (2x + 3)
(2x + 3)² = (2x) ² + 2*2x*3 + (3²) = 4x² + 12x + 9
(4x² + 12x + 9) * (2x + 3) = 4x²*2x + 4x²*3 + 12x*2x + 12x*3 + 9*2x + 9*3 =
8x³ + 12x² + 24x² + 36x + 18x + 27 = 8x³ + 36x² + 54x + 27
rule of thumb
"The cube of the first term plus three times the square of the first term times the second term plus three times the first term times the square of the second term plus the cube of the second term."
(x + 3)³ = (x) ³ + 3*(x) ²*3 + 3*x*(3)² + (3)³ = x³ + 9x² + 27x + 27
(2b + 2)³ = (2b) ³ + 3*(2b) ²*2 + 3*2b*(2)² + (2)³ = 8b³ + 24b² + 24b + 8
Cube of Difference
The difference cube can be developed according to the solving principles of the sum cube. The only change to be made is regarding the use of the negative sign.
rule of thumb
"The cube of the first term minus three times the square of the first term times the second term plus three times the first term times the square of the second term minus the cube of the second term."
(x – 3)³ = (x) ³ – 3*(x) ²*3 + 3*x*(3)² – (3)³ = x³ - 9x² + 27x - 27
(2b – 2)³ = (2b) ³ – 3*(2b) ²*2 + 3*2b*(2)² – (2)³ = 8b³ - 24b² + 24b - 8
by Mark Noah
Graduated in Mathematics
Brazil School Team
Notable products - Math - Brazil School
Source: Brazil School - https://brasilescola.uol.com.br/matematica/cubo-soma-cubo-diferenca.htm
