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شائع الجبر >

expand (x+y)^{16}

الحلّ

expand

(x + y)^{16}

الحلّ

x^{16} + 16 x^{15} y + 120 x^{14} y^{2} + 560 x^{13} y^{3} + 1820 x^{12} y^{4} + 4368 x^{11} y^{5} + 8008 x^{10} y^{6} + 11440 x^{9} y^{7} + 12870 x^{8} y^{8} + 11440 x^{7} y^{9} + 8008 x^{6} y^{10} + 4368 x^{5} y^{11} + 1820 x^{4} y^{12} + 560 x^{3} y^{13} + 120 x^{2} y^{14} + 16 x y^{15} + y^{16}

خطوات الحلّ

(x + y)^{16}

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

:فعّل نظريّة ذات الحدّين لنيوتن

a = x, b = y

= i = 0 \sum 16 (i 16) x^{(16 - i)} y^{i}

وسّع حاصل الجمع

= \frac{16 !}{0 ! ( 16 - 0 ) !} x^{16} y^{0} + \frac{16 !}{1 ! ( 16 - 1 ) !} x^{15} y^{1} + \frac{16 !}{2 ! ( 16 - 2 ) !} x^{14} y^{2} + \frac{16 !}{3 ! ( 16 - 3 ) !} x^{13} y^{3} + \frac{16 !}{4 ! ( 16 - 4 ) !} x^{12} y^{4} + \frac{16 !}{5 ! ( 16 - 5 ) !} x^{11} y^{5} + \frac{16 !}{6 ! ( 16 - 6 ) !} x^{10} y^{6} + \frac{16 !}{7 ! ( 16 - 7 ) !} x^{9} y^{7} + \frac{16 !}{8 ! ( 16 - 8 ) !} x^{8} y^{8} + \frac{16 !}{9 ! ( 16 - 9 ) !} x^{7} y^{9} + \frac{16 !}{10 ! ( 16 - 10 ) !} x^{6} y^{10} + \frac{16 !}{11 ! ( 16 - 11 ) !} x^{5} y^{11} + \frac{16 !}{12 ! ( 16 - 12 ) !} x^{4} y^{12} + \frac{16 !}{13 ! ( 16 - 13 ) !} x^{3} y^{13} + \frac{16 !}{14 ! ( 16 - 14 ) !} x^{2} y^{14} + \frac{16 !}{15 ! ( 16 - 15 ) !} x^{1} y^{15} + \frac{16 !}{16 ! ( 16 - 16 ) !} x^{0} y^{16}

\frac{16 !}{0 ! ( 16 - 0 ) !} x^{16} y^{0}

بسّط:

x^{16}

\frac{16 !}{1 ! ( 16 - 1 ) !} x^{15} y^{1}

بسّط:

16 x^{15} y

\frac{16 !}{2 ! ( 16 - 2 ) !} x^{14} y^{2}

بسّط:

120 x^{14} y^{2}

\frac{16 !}{3 ! ( 16 - 3 ) !} x^{13} y^{3}

بسّط:

560 x^{13} y^{3}

\frac{16 !}{4 ! ( 16 - 4 ) !} x^{12} y^{4}

بسّط:

1820 x^{12} y^{4}

\frac{16 !}{5 ! ( 16 - 5 ) !} x^{11} y^{5}

بسّط:

4368 x^{11} y^{5}

\frac{16 !}{6 ! ( 16 - 6 ) !} x^{10} y^{6}

بسّط:

8008 x^{10} y^{6}

\frac{16 !}{7 ! ( 16 - 7 ) !} x^{9} y^{7}

بسّط:

11440 x^{9} y^{7}

\frac{16 !}{8 ! ( 16 - 8 ) !} x^{8} y^{8}

بسّط:

12870 x^{8} y^{8}

\frac{16 !}{9 ! ( 16 - 9 ) !} x^{7} y^{9}

بسّط:

11440 x^{7} y^{9}

\frac{16 !}{10 ! ( 16 - 10 ) !} x^{6} y^{10}

بسّط:

8008 x^{6} y^{10}

\frac{16 !}{11 ! ( 16 - 11 ) !} x^{5} y^{11}

بسّط:

4368 x^{5} y^{11}

\frac{16 !}{12 ! ( 16 - 12 ) !} x^{4} y^{12}

بسّط:

1820 x^{4} y^{12}

\frac{16 !}{13 ! ( 16 - 13 ) !} x^{3} y^{13}

بسّط:

560 x^{3} y^{13}

\frac{16 !}{14 ! ( 16 - 14 ) !} x^{2} y^{14}

بسّط:

120 x^{2} y^{14}

\frac{16 !}{15 ! ( 16 - 15 ) !} x^{1} y^{15}

بسّط:

16 x y^{15}

\frac{16 !}{16 ! ( 16 - 16 ) !} x^{0} y^{16}

بسّط:

y^{16}

= x^{16} + 16 x^{15} y + 120 x^{14} y^{2} + 560 x^{13} y^{3} + 1820 x^{12} y^{4} + 4368 x^{11} y^{5} + 8008 x^{10} y^{6} + 11440 x^{9} y^{7} + 12870 x^{8} y^{8} + 11440 x^{7} y^{9} + 8008 x^{6} y^{10} + 4368 x^{5} y^{11} + 1820 x^{4} y^{12} + 560 x^{3} y^{13} + 120 x^{2} y^{14} + 16 x y^{15} + y^{16}

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