Notable products are binomial multiplications that respect a standard form of resolution. The square of the sum of two terms (a + b) ², the square of the difference of two terms (a – b) ², the cube of the sum of two terms (a + b) ³ and the cube of the difference of two terms (a – b) ³ are the main notable products within the Math. Another product involving multiplications of the type (x + a) * (x + b) is also known, as it generates trinomials considered not perfect.
Perfect trinomials are connected to the square of the sum of two terms and the square of the difference of two terms. Take a look at some examples:
x² + 6x + 9 = (x + 3)² = (x + 3) *(x + 3)
x² + 16x + 64 = (x + 8)² = (x + 8) * (x + 8)
x² – 24x + 144 = (x – 12)² = (x – 12) * (x – 12)
x² – 20x + 100 = (x – 10)² = (x – 10) * (x – 10)
The non-perfect trinomials are linked to the multiplications (x + a) * (x + b) and are also called trinomials: sum and product. Watch:
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(x + a) * (x + b) → x² + b*x + a*x + a*b → x² + x * (b + a) +a*b
The trinomial result of multiplication (x + a) * (x + b) can be written in the form
x² + Sx + P, where S is the sum of a + b and P the product of a and b.
(x + 3) * (x + 6) = x² + (3 + 6)x + 6 * 3 = x² + 9x + 18
(x – 4) * (x + 8) = x² + (–4 + 8)x + (–4) * 8 = x² + 4x – 32
(x – 12) * (x – 5) = x² + (–12 –5)x + (–12) * (–5) = x² - 17x + 60
(x + 7) * (x – 9) = x² + (7 – 9)x + (– 9) * 7 = x² -2x - 63
by Mark Noah
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/matematica/produto-tipo-x--x-b.htm
