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Popolare Algebra >

espandere (x+4y)^8

Soluzione

espandere

(x + 4 y)^{8}

Soluzione

x^{8} + 32 x^{7} y + 448 x^{6} y^{2} + 3584 x^{5} y^{3} + 17920 x^{4} y^{4} + 57344 x^{3} y^{5} + 114688 x^{2} y^{6} + 131072 x y^{7} + 65536 y^{8}

Fasi della soluzione

(x + 4 y)^{8}

Applicare il teorema binomiale:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = x, b = 4 y

= i = 0 \sum 8 (i 8) x^{(8 - i)} (4 y)^{i}

Espandere la sommatoria

= \frac{8 !}{0 ! ( 8 - 0 ) !} x^{8} (4 y)^{0} + \frac{8 !}{1 ! ( 8 - 1 ) !} x^{7} (4 y)^{1} + \frac{8 !}{2 ! ( 8 - 2 ) !} x^{6} (4 y)^{2} + \frac{8 !}{3 ! ( 8 - 3 ) !} x^{5} (4 y)^{3} + \frac{8 !}{4 ! ( 8 - 4 ) !} x^{4} (4 y)^{4} + \frac{8 !}{5 ! ( 8 - 5 ) !} x^{3} (4 y)^{5} + \frac{8 !}{6 ! ( 8 - 6 ) !} x^{2} (4 y)^{6} + \frac{8 !}{7 ! ( 8 - 7 ) !} x^{1} (4 y)^{7} + \frac{8 !}{8 ! ( 8 - 8 ) !} x^{0} (4 y)^{8}

Semplificare

\frac{8 !}{0 ! ( 8 - 0 ) !} x^{8} (4 y)^{0} : x^{8}

Semplificare

\frac{8 !}{1 ! ( 8 - 1 ) !} x^{7} (4 y)^{1} : 32 x^{7} y

Semplificare

\frac{8 !}{2 ! ( 8 - 2 ) !} x^{6} (4 y)^{2} : 448 x^{6} y^{2}

Semplificare

\frac{8 !}{3 ! ( 8 - 3 ) !} x^{5} (4 y)^{3} : 3584 x^{5} y^{3}

Semplificare

\frac{8 !}{4 ! ( 8 - 4 ) !} x^{4} (4 y)^{4} : 17920 x^{4} y^{4}

Semplificare

\frac{8 !}{5 ! ( 8 - 5 ) !} x^{3} (4 y)^{5} : 57344 x^{3} y^{5}

Semplificare

\frac{8 !}{6 ! ( 8 - 6 ) !} x^{2} (4 y)^{6} : 114688 x^{2} y^{6}

Semplificare

\frac{8 !}{7 ! ( 8 - 7 ) !} x^{1} (4 y)^{7} : 131072 x y^{7}

Semplificare

\frac{8 !}{8 ! ( 8 - 8 ) !} x^{0} (4 y)^{8} : 65536 y^{8}

= x^{8} + 32 x^{7} y + 448 x^{6} y^{2} + 3584 x^{5} y^{3} + 17920 x^{4} y^{4} + 57344 x^{3} y^{5} + 114688 x^{2} y^{6} + 131072 x y^{7} + 65536 y^{8}

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