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Beliebt Algebra >

erweitern (x+sqrt(x^2+1))^{10}

Lösung

erweitern

(x + x^{2} + 1)^{10}

Lösung

512 x^{10} + 10 x^{9} x^{2} + 1 + 1280 x^{8} + 120 x^{7} x^{2} + 1 (x^{2} + 1) + 1120 x^{6} + 252 x^{5} (x^{2} + 1)^{2} x^{2} + 1 + 400 x^{4} + 120 x^{3} (x^{2} + 1)^{3} x^{2} + 1 + 50 x^{2} + 10 x (x^{2} + 1)^{4} x^{2} + 1 + 1

Schritte zur Lösung

(x + x^{2} + 1)^{10}

Wende Binomialsetz an:

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

a = x, b = x^{2} + 1

= i = 0 \sum 10 (i 10) x^{(10 - i)} (x^{2} + 1)^{i}

Erweitere Summation

= \frac{10 !}{0 ! ( 10 - 0 ) !} x^{10} (x^{2} + 1)^{0} + \frac{10 !}{1 ! ( 10 - 1 ) !} x^{9} (x^{2} + 1)^{1} + \frac{10 !}{2 ! ( 10 - 2 ) !} x^{8} (x^{2} + 1)^{2} + \frac{10 !}{3 ! ( 10 - 3 ) !} x^{7} (x^{2} + 1)^{3} + \frac{10 !}{4 ! ( 10 - 4 ) !} x^{6} (x^{2} + 1)^{4} + \frac{10 !}{5 ! ( 10 - 5 ) !} x^{5} (x^{2} + 1)^{5} + \frac{10 !}{6 ! ( 10 - 6 ) !} x^{4} (x^{2} + 1)^{6} + \frac{10 !}{7 ! ( 10 - 7 ) !} x^{3} (x^{2} + 1)^{7} + \frac{10 !}{8 ! ( 10 - 8 ) !} x^{2} (x^{2} + 1)^{8} + \frac{10 !}{9 ! ( 10 - 9 ) !} x^{1} (x^{2} + 1)^{9} + \frac{10 !}{10 ! ( 10 - 10 ) !} x^{0} (x^{2} + 1)^{10}

Vereinfache

\frac{10 !}{0 ! ( 10 - 0 ) !} x^{10} (x^{2} + 1)^{0} : x^{10}

Vereinfache

\frac{10 !}{1 ! ( 10 - 1 ) !} x^{9} (x^{2} + 1)^{1} : 10 x^{9} x^{2} + 1

Vereinfache

\frac{10 !}{2 ! ( 10 - 2 ) !} x^{8} (x^{2} + 1)^{2} : 45 x^{8} (x^{2} + 1)

Vereinfache

\frac{10 !}{3 ! ( 10 - 3 ) !} x^{7} (x^{2} + 1)^{3} : 120 x^{7} (x^{2} + 1)^{3}

Vereinfache

\frac{10 !}{4 ! ( 10 - 4 ) !} x^{6} (x^{2} + 1)^{4} : 210 x^{6} (x^{2} + 1)^{2}

Vereinfache

\frac{10 !}{5 ! ( 10 - 5 ) !} x^{5} (x^{2} + 1)^{5} : 252 x^{5} (x^{2} + 1)^{5}

Vereinfache

\frac{10 !}{6 ! ( 10 - 6 ) !} x^{4} (x^{2} + 1)^{6} : 210 x^{4} (x^{2} + 1)^{3}

Vereinfache

\frac{10 !}{7 ! ( 10 - 7 ) !} x^{3} (x^{2} + 1)^{7} : 120 x^{3} (x^{2} + 1)^{7}

Vereinfache

\frac{10 !}{8 ! ( 10 - 8 ) !} x^{2} (x^{2} + 1)^{8} : 45 x^{2} (x^{2} + 1)^{4}

Vereinfache

\frac{10 !}{9 ! ( 10 - 9 ) !} x^{1} (x^{2} + 1)^{9} : 10 x (x^{2} + 1)^{9}

Vereinfache

\frac{10 !}{10 ! ( 10 - 10 ) !} x^{0} (x^{2} + 1)^{10} : (x^{2} + 1)^{5}

= x^{10} + 10 x^{9} x^{2} + 1 + 45 x^{8} (x^{2} + 1) + 120 x^{7} (x^{2} + 1)^{3} + 210 x^{6} (x^{2} + 1)^{2} + 252 x^{5} (x^{2} + 1)^{5} + 210 x^{4} (x^{2} + 1)^{3} + 120 x^{3} (x^{2} + 1)^{7} + 45 x^{2} (x^{2} + 1)^{4} + 10 x (x^{2} + 1)^{9} + (x^{2} + 1)^{5}

Schreibe

45 x^{8} (x^{2} + 1)

um:

45 x^{10} + 45 x^{8}

= x^{10} + 10 x^{9} x^{2} + 1 + 45 x^{10} + 45 x^{8} + 120 x^{7} (x^{2} + 1)^{3} + 210 x^{6} (x^{2} + 1)^{2} + 252 x^{5} (x^{2} + 1)^{5} + 210 x^{4} (x^{2} + 1)^{3} + 120 x^{3} (x^{2} + 1)^{7} + 45 x^{2} (x^{2} + 1)^{4} + 10 x (x^{2} + 1)^{9} + (x^{2} + 1)^{5}

Schreibe

210 x^{6} (x^{2} + 1)^{2}

um:

210 x^{10} + 420 x^{8} + 210 x^{6}

= x^{10} + 10 x^{9} x^{2} + 1 + 45 x^{10} + 45 x^{8} + 120 x^{7} (x^{2} + 1)^{3} + 210 x^{10} + 420 x^{8} + 210 x^{6} + 252 x^{5} (x^{2} + 1)^{5} + 210 x^{4} (x^{2} + 1)^{3} + 120 x^{3} (x^{2} + 1)^{7} + 45 x^{2} (x^{2} + 1)^{4} + 10 x (x^{2} + 1)^{9} + (x^{2} + 1)^{5}

Schreibe

210 x^{4} (x^{2} + 1)^{3}

um:

210 x^{10} + 630 x^{8} + 630 x^{6} + 210 x^{4}

= x^{10} + 10 x^{9} x^{2} + 1 + 45 x^{10} + 45 x^{8} + 120 x^{7} (x^{2} + 1)^{3} + 210 x^{10} + 420 x^{8} + 210 x^{6} + 252 x^{5} (x^{2} + 1)^{5} + 210 x^{10} + 630 x^{8} + 630 x^{6} + 210 x^{4} + 120 x^{3} (x^{2} + 1)^{7} + 45 x^{2} (x^{2} + 1)^{4} + 10 x (x^{2} + 1)^{9} + (x^{2} + 1)^{5}

Schreibe

45 x^{2} (x^{2} + 1)^{4}

um:

45 x^{10} + 180 x^{8} + 270 x^{6} + 180 x^{4} + 45 x^{2}

= x^{10} + 10 x^{9} x^{2} + 1 + 45 x^{10} + 45 x^{8} + 120 x^{7} (x^{2} + 1)^{3} + 210 x^{10} + 420 x^{8} + 210 x^{6} + 252 x^{5} (x^{2} + 1)^{5} + 210 x^{10} + 630 x^{8} + 630 x^{6} + 210 x^{4} + 120 x^{3} (x^{2} + 1)^{7} + 45 x^{10} + 180 x^{8} + 270 x^{6} + 180 x^{4} + 45 x^{2} + 10 x (x^{2} + 1)^{9} + (x^{2} + 1)^{5}

Schreibe

(x^{2} + 1)^{5}

um:

x^{10} + 5 x^{8} + 10 x^{6} + 10 x^{4} + 5 x^{2} + 1

= x^{10} + 10 x^{9} x^{2} + 1 + 45 x^{10} + 45 x^{8} + 120 x^{7} (x^{2} + 1)^{3} + 210 x^{10} + 420 x^{8} + 210 x^{6} + 252 x^{5} (x^{2} + 1)^{5} + 210 x^{10} + 630 x^{8} + 630 x^{6} + 210 x^{4} + 120 x^{3} (x^{2} + 1)^{7} + 45 x^{10} + 180 x^{8} + 270 x^{6} + 180 x^{4} + 45 x^{2} + 10 x (x^{2} + 1)^{9} + x^{10} + 5 x^{8} + 10 x^{6} + 10 x^{4} + 5 x^{2} + 1

Vereinfache

x^{10} + 10 x^{9} x^{2} + 1 + 45 x^{10} + 45 x^{8} + 120 x^{7} (x^{2} + 1)^{3} + 210 x^{10} + 420 x^{8} + 210 x^{6} + 252 x^{5} (x^{2} + 1)^{5} + 210 x^{10} + 630 x^{8} + 630 x^{6} + 210 x^{4} + 120 x^{3} (x^{2} + 1)^{7} + 45 x^{10} + 180 x^{8} + 270 x^{6} + 180 x^{4} + 45 x^{2} + 10 x (x^{2} + 1)^{9} + x^{10} + 5 x^{8} + 10 x^{6} + 10 x^{4} + 5 x^{2} + 1 : 512 x^{10} + 10 x^{9} x^{2} + 1 + 1280 x^{8} + 120 x^{7} x^{2} + 1 (x^{2} + 1) + 1120 x^{6} + 252 x^{5} (x^{2} + 1)^{2} x^{2} + 1 + 400 x^{4} + 120 x^{3} (x^{2} + 1)^{3} x^{2} + 1 + 50 x^{2} + 10 x (x^{2} + 1)^{4} x^{2} + 1 + 1

= 512 x^{10} + 10 x^{9} x^{2} + 1 + 1280 x^{8} + 120 x^{7} x^{2} + 1 (x^{2} + 1) + 1120 x^{6} + 252 x^{5} (x^{2} + 1)^{2} x^{2} + 1 + 400 x^{4} + 120 x^{3} (x^{2} + 1)^{3} x^{2} + 1 + 50 x^{2} + 10 x (x^{2} + 1)^{4} x^{2} + 1 + 1

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