expand ((156+x)^{10})/(10)

פתרון

expand

\frac{( 156 + x ) ^{10}}{10}

\frac{x ^{10}}{10} + 156 x^{9} + 109512 x^{8} + 45556992 x^{7} + 12437058816 x^{6} + \frac{15 6 ^{5} \cdot 126 x ^{5}}{5} + 15 6^{6} \cdot 21 x^{4} + 15 6^{7} \cdot 12 x^{3} + 1.57837 E 18 x^{2} + 15 6^{9} x + \frac{1 3 ^{10} \cdot 30958682112}{5}

צעדי פתרון

\frac{( 156 + x ) ^{10}}{10}

(156 + x)^{10} = 15 6^{10} + 15 6^{9} \cdot 10 x + 15 6^{8} \cdot 45 x^{2} + 15 6^{7} \cdot 120 x^{3} + 15 6^{6} \cdot 210 x^{4} + 15 6^{5} \cdot 252 x^{5} + 124370588160 x^{6} + 455569920 x^{7} + 1095120 x^{8} + 1560 x^{9} + x^{10}

= \frac{15 6 ^{10} + 15 6 ^{9} \cdot 10 x + 15 6 ^{8} \cdot 45 x ^{2} + 15 6 ^{7} \cdot 120 x ^{3} + 15 6 ^{6} \cdot 210 x ^{4} + 15 6 ^{5} \cdot 252 x ^{5} + 124370588160 x ^{6} + 455569920 x ^{7} + 1095120 x ^{8} + 1560 x ^{9} + x ^{10}}{10}

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

: השתמש בתכונת השברים הבאה

\frac{15 6 ^{10} + 15 6 ^{9} \cdot 10 x + 15 6 ^{8} \cdot 45 x ^{2} + 15 6 ^{7} \cdot 120 x ^{3} + 15 6 ^{6} \cdot 210 x ^{4} + 15 6 ^{5} \cdot 252 x ^{5} + 124370588160 x ^{6} + 455569920 x ^{7} + 1095120 x ^{8} + 1560 x ^{9} + x ^{10}}{10} = \frac{15 6 ^{10}}{10} + \frac{15 6 ^{9} \cdot 10 x}{10} + \frac{15 6 ^{8} \cdot 45 x ^{2}}{10} + \frac{15 6 ^{7} \cdot 120 x ^{3}}{10} + \frac{15 6 ^{6} \cdot 210 x ^{4}}{10} + \frac{15 6 ^{5} \cdot 252 x ^{5}}{10} + \frac{124370588160 x ^{6}}{10} + \frac{455569920 x ^{7}}{10} + \frac{1095120 x ^{8}}{10} + \frac{1560 x ^{9}}{10} + \frac{x ^{10}}{10}

= \frac{15 6 ^{10}}{10} + \frac{15 6 ^{9} \cdot 10 x}{10} + \frac{15 6 ^{8} \cdot 45 x ^{2}}{10} + \frac{15 6 ^{7} \cdot 120 x ^{3}}{10} + \frac{15 6 ^{6} \cdot 210 x ^{4}}{10} + \frac{15 6 ^{5} \cdot 252 x ^{5}}{10} + \frac{124370588160 x ^{6}}{10} + \frac{455569920 x ^{7}}{10} + \frac{1095120 x ^{8}}{10} + \frac{1560 x ^{9}}{10} + \frac{x ^{10}}{10}

\frac{15 6 ^{10}}{10} = \frac{1 3 ^{10} \cdot 30958682112}{5}

\frac{15 6 ^{9} \cdot 10 x}{10} = 15 6^{9} x

\frac{15 6 ^{8} \cdot 45 x ^{2}}{10} = 1.57837 E 18 x^{2}

\frac{15 6 ^{7} \cdot 120 x ^{3}}{10} = 15 6^{7} \cdot 12 x^{3}

\frac{15 6 ^{6} \cdot 210 x ^{4}}{10} = 15 6^{6} \cdot 21 x^{4}

\frac{15 6 ^{5} \cdot 252 x ^{5}}{10} = \frac{15 6 ^{5} \cdot 126 x ^{5}}{5}

\frac{124370588160 x ^{6}}{10} = 12437058816 x^{6}

\frac{455569920 x ^{7}}{10} = 45556992 x^{7}

\frac{1095120 x ^{8}}{10} = 109512 x^{8}

\frac{1560 x ^{9}}{10} = 156 x^{9}

= \frac{1 3 ^{10} \cdot 30958682112}{5} + 15 6^{9} x + 1.57837 E 18 x^{2} + 15 6^{7} \cdot 12 x^{3} + 15 6^{6} \cdot 21 x^{4} + \frac{15 6 ^{5} \cdot 126 x ^{5}}{5} + 12437058816 x^{6} + 45556992 x^{7} + 109512 x^{8} + 156 x^{9} + \frac{x ^{10}}{10}

שכתב בצורה סטנדרטית

= \frac{x ^{10}}{10} + 156 x^{9} + 109512 x^{8} + 45556992 x^{7} + 12437058816 x^{6} + \frac{15 6 ^{5} \cdot 126 x ^{5}}{5} + 15 6^{6} \cdot 21 x^{4} + 15 6^{7} \cdot 12 x^{3} + 1.57837 E 18 x^{2} + 15 6^{9} x + \frac{1 3 ^{10} \cdot 30958682112}{5}

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