Expresia devine
[tex]E(x)=(5x+3)^{2}-(3x+4)^{2} =[/tex]
[tex]\boxed{\boldsymbol{(a - b)(a + b) = a^{2} - b^{2}}}[/tex]
[tex] = (5x + 3 - 3x - 4)(5x + 3 + 3x + 4) \\ [/tex]
[tex]= (2x - 1)(8x + 7)[/tex]
______
[tex]E(n) = (2n - 1)(8n + 7)[/tex]
Dacă n nu este divizibil cu 3, atunci este de forma 3k+1 sau 3k+2
[tex]E(3k + 1) = [2 \times (3k + 1) - 1] \times [8 \times (3k + 1) + 7] = \\ [/tex]
[tex]= (6k + 2 - 1)(24k + 8 + 7) \\ [/tex]
[tex]= (6k + 1)(24k + 15)[/tex]
[tex]= 3 \times (6k + 1)(8k + 5)\in \mathcal {M}_{3} \\ [/tex]
[tex]E(3k + 2) = [2 \times (3k + 2) - 1] \times [8 \times (3k + 2) + 7] = \\ [/tex]
[tex]= (6k + 4 - 1)(24k + 16 + 7) \\ [/tex]
[tex]= (6k + 3)(24k + 23)[/tex]
[tex]= 3 \times (2k + 1)(24k + 23)\in \mathcal {M}_{3} \\ [/tex]
q.e.d.
______
brainly.ro/tema/10776449
