Question

calcule o 3 termo do
desenvolvimento de (x+2)5​

Answer 1

Usando o Binômio de Newton:

(x+2)⁵ = $\left[\begin{array}{cc}5&\\0&\\\end{array}\right]$  x⁵.2⁰ + $\left[\begin{array}{cc}5&\\1&\\\end{array}\right]$  x⁴.2¹+$\left[\begin{array}{cc}5&\\2&\\\end{array}\right]$ x³.2² + $\left[\begin{array}{cc}5&\\3&\\\end{array}\right]$ x².2³+$\left[\begin{array}{cc}5&\\4&\\\end{array}\right]$ x¹.2⁴

            +$\left[\begin{array}{cc}5&\\5&\\\end{array}\right]$ x⁰.2⁵

(x+2)⁵ =  1. x⁵.1 + 5. x⁴.2 + 10. x³.4 + 10. x².8 + 5. x.16 + 1. 1. 32

(x+2)⁵ = x⁵ + 10x⁴ + 40x³ + 80x² + 80x + 32

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Podemos resolver também usando "O quadrado da soma entre dois termos" :

(x+2)² . (x+2)² . (x+2) =

(x² + 4x + 4) (x² + 4x + 4) . (x +2) =

[x² (x² + 4x + 4) + 4x (x² + 4x + 4) + 4 (x² + 4x + 4)] . (x+2) =

[x⁴ + 4x³ + 4x² + 4x³ + 16x² + 16x + 4x² + 16x + 16] . (x+2) =

[x⁴ + 8x³ + 24x² +32x + 16] . (x+2) =

x⁵ + 8x⁴ + 24x³ + 32x² + 16x + 2x⁴ + 16x³ + 48x²+64x + 32 =

x⁵ + 10x⁴ + 40x³ + 80x² + 80x + 32

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Nos anexos, o Binômio de Newton e os Produtos Notáveis.