An interesting situation involving algebraic expressions is presented as follows:
(a + b)(a – b), being called the Product of the Sum of the Difference, which can be resolved through the distributive property of multiplication or through a practical rule. This expression can be considered a remarkable product, due to the regular characteristic presented in the resolution of similar situations.
Applying the distributive property in solving the expression (a + b)(a – b).
(a + b)(a - b) = a*a - a*b + b*a - b*b = a² - b²
Note that the terms – ab and + ba are opposites, so they cancel each other out.
(2x + 4) (2x – 4) = 2x*2x – 2x*4 + 4*2x – 4*4 = 4x² – 8x + 8x – 16 = 4x² - 16
(7x + 6)(7x – 6) = 7x*7x – 7x*6 + 6*7x – 6*6 = 49x² – 42x + 42x – 36 = 49x² - 36
(10x³ – 12)(10x³ + 12) = 10x³*10x³ + 10x³*12 – 12*10x³ –12*12 = 100x6 + 120x³ – 120x³ – 144 = 100x6 – 144
(20z + 10x)(20z – 10x) = 20z*20z – 20z*10x + 10x*20z – 10x*10x = 400z² – 200zx + 200xz – 100x² = 400z² - 100x²
Applying the rule of thumb
The application of the practical rule takes place through the following situation: "the first term squared minus the second term squared"
(4x + 7)(4x - 7) = (4x) ² - (7)² = 16x² - 49
(12x + 8)(12x - 8) = (12x) ² - (8)² = 144x² - 64
(11x² - 5x) (11x² + 5x) = (11x²)² - (5x) ² = 121x4 – 25x²
(20b – 30)(20b + 30) = (20b) ² – (30)² = 400b² - 900
by Mark Noah
Graduated in Mathematics
Brazil School Team
Notable products - Math - Brazil School
Source: Brazil School - https://brasilescola.uol.com.br/matematica/produto-soma-pela-diferenca.htm