Question

Escreva todos os termos da expansão dos binômios a seguir:
A) (X+Y) ^5
B) (A-1)^6

Answer 1

Olá, boa tarde.

a)

$(x + y) {}^{5} = \binom{5}{0} .x {}^{5} + \binom{5}{1} .x {}^{4} .y + \binom{5}{2} .x {}^{3} .y {}^{2} + \binom{5}{3} .x {}^{2} .y {}^{3} + \binom{5}{4} x.y {}^{4} + \binom{5}{5} .y {}^{5} \\ (x + y) {}^{5} = 1.x {}^{5} + 5.x {}^{4} .y + \binom{5}{2} x {}^{3} .y {}^{2} + \binom{5}{3} .x {}^{2} .y {}^{3} + \binom{5}{4} x.y {}^{4} + 1.y {}^{5} \\ \\ \binom{5}{2} = \frac{5}{2.(5 - 2)} = \frac{5}{2.3} = \frac{5.4.3}{2.3} = \frac{5.4}{2} = \frac{20}{2} = 10 \\ \binom{5}{3} = \frac{5}{3.(5 - 3)} = \frac{5}{3.2} = 10 \\ \binom{5}{4} = 5 \\ \\ (x + y) {}^{5} = x {}^{5} + 5x {}^{4} y + 10x {}^{3} y {}^{2} + 10x {}^{2} y {}^{3} + 5xy {}^{4} + y {}^{5}$

b)

$(a - 1) {}^{6} = \binom{6}{0} .a {}^{6} - \binom{6}{1} .a {}^{5} + \binom{6}{2} .a {}^{4} - \binom{6}{3} .a {}^{3} + \binom{6}{4} .a {}^{2} - \binom{6}{5} .a \: + \binom{6}{6} .1 \\ (a - 1) {}^{6} = 1.a {}^{6} - 6.a {}^{5} + \binom{6}{2} .a {}^{4} - \binom{6}{3} .a {}^{3} + \binom{6}{4} .a {}^{2} - 6.a \: + 1.1 \\ \\ \binom{6}{2} = \frac{6}{2.(6 - 2)} = \frac{6}{2.4} = \frac{6.5.4}{2.4} = \frac{6.5}{2} = \frac{30}{2} = 15 \\ \binom{6}{3} = \frac{6}{3.(6 - 3)} = \frac{6}{3.3} = \frac{6.5.4.3}{3.3} = \frac{6.5.4}{3.2.1} = \frac{120}{6} = 20 \\ \binom{6}{4} = \frac{6}{4.(6 - 4)} = \frac{6}{4.2} = 15 \\ \\ (a - 1) {}^{6} = a {}^{6} - 6a {}^{5} + 15a {}^{4} - 20a {}^{3} + 15a {}^{2} - 6a + 1$

Espero ter ajudado

Bons estudos ♥️