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شائع الجبر >

expand (2x+3)^{12}

الحلّ

expand

(2 x + 3)^{12}

الحلّ

4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441

خطوات الحلّ

(2 x + 3)^{12}

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

:فعّل نظريّة ذات الحدّين لنيوتن

a = 2 x, b = 3

= i = 0 \sum 12 (i 12) (2 x)^{(12 - i)} \cdot 3^{i}

وسّع حاصل الجمع

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

\frac{12 !}{0 ! ( 12 - 0 ) !} (2 x)^{12} \cdot 3^{0}

بسّط:

4096 x^{12}

\frac{12 !}{1 ! ( 12 - 1 ) !} (2 x)^{11} \cdot 3^{1}

بسّط:

73728 x^{11}

\frac{12 !}{2 ! ( 12 - 2 ) !} (2 x)^{10} \cdot 3^{2}

بسّط:

608256 x^{10}

\frac{12 !}{3 ! ( 12 - 3 ) !} (2 x)^{9} \cdot 3^{3}

بسّط:

3041280 x^{9}

\frac{12 !}{4 ! ( 12 - 4 ) !} (2 x)^{8} \cdot 3^{4}

بسّط:

10264320 x^{8}

\frac{12 !}{5 ! ( 12 - 5 ) !} (2 x)^{7} \cdot 3^{5}

بسّط:

24634368 x^{7}

\frac{12 !}{6 ! ( 12 - 6 ) !} (2 x)^{6} \cdot 3^{6}

بسّط:

43110144 x^{6}

\frac{12 !}{7 ! ( 12 - 7 ) !} (2 x)^{5} \cdot 3^{7}

بسّط:

55427328 x^{5}

\frac{12 !}{8 ! ( 12 - 8 ) !} (2 x)^{4} \cdot 3^{8}

بسّط:

51963120 x^{4}

\frac{12 !}{9 ! ( 12 - 9 ) !} (2 x)^{3} \cdot 3^{9}

بسّط:

34642080 x^{3}

\frac{12 !}{10 ! ( 12 - 10 ) !} (2 x)^{2} \cdot 3^{10}

بسّط:

15588936 x^{2}

\frac{12 !}{11 ! ( 12 - 11 ) !} (2 x)^{1} \cdot 3^{11}

بسّط:

4251528 x

\frac{12 !}{12 ! ( 12 - 12 ) !} (2 x)^{0} \cdot 3^{12}

بسّط:

531441

= 4096 x^{12} + 73728 x^{11} + 608256 x^{10} + 3041280 x^{9} + 10264320 x^{8} + 24634368 x^{7} + 43110144 x^{6} + 55427328 x^{5} + 51963120 x^{4} + 34642080 x^{3} + 15588936 x^{2} + 4251528 x + 531441

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