espandere (bc+de)^6

Soluzione

espandere

(b c + d e)^{6}

b^{6} c^{6} + 6 d e b^{5} c^{5} + 15 d e^{2} b^{4} c^{4} + 20 d e^{3} b^{3} c^{3} + 15 d e^{4} b^{2} c^{2} + 6 d e^{5} b c + d e^{6}

Fasi della soluzione

(b c + d e)^{6}

Applicare il teorema binomiale:

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

a = b c, b = d e

= i = 0 \sum 6 (i 6) (b c)^{(6 - i)} d e^{i}

Espandere la sommatoria

= \frac{6 !}{0 ! ( 6 - 0 ) !} (b c)^{6} d e^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} (b c)^{5} d e^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} (b c)^{4} d e^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} (b c)^{3} d e^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} (b c)^{2} d e^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} (b c)^{1} d e^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} (b c)^{0} d e^{6}

Semplifica

\frac{6 !}{0 ! ( 6 - 0 ) !} (b c)^{6} d e^{0} : b^{6} c^{6}

Semplifica

\frac{6 !}{1 ! ( 6 - 1 ) !} (b c)^{5} d e^{1} : 6 d e b^{5} c^{5}

Semplifica

\frac{6 !}{2 ! ( 6 - 2 ) !} (b c)^{4} d e^{2} : 15 d e^{2} b^{4} c^{4}

Semplifica

\frac{6 !}{3 ! ( 6 - 3 ) !} (b c)^{3} d e^{3} : 20 d e^{3} b^{3} c^{3}

Semplifica

\frac{6 !}{4 ! ( 6 - 4 ) !} (b c)^{2} d e^{4} : 15 d e^{4} b^{2} c^{2}

Semplifica

\frac{6 !}{5 ! ( 6 - 5 ) !} (b c)^{1} d e^{5} : 6 d e^{5} b c

Semplifica

\frac{6 !}{6 ! ( 6 - 6 ) !} (b c)^{0} d e^{6} : d e^{6}

= b^{6} c^{6} + 6 d e b^{5} c^{5} + 15 d e^{2} b^{4} c^{4} + 20 d e^{3} b^{3} c^{3} + 15 d e^{4} b^{2} c^{2} + 6 d e^{5} b c + d e^{6}

e x p an d (1 - x) y^{''} (x) + x y^{'} (x) - y

e x p an d (f^{'} (t) sin (t) + f (t) cos (t))^{2}

e x p an d (sin (\frac{x}{2}) + cos (\frac{x}{2}))^{2} - sin (x)

s im pl i f y (x + 3 - 23) (x + 3 + 23)

s im pl i f y (z - \frac{1}{2}) (z - \frac{1}{5})