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Popolare Algebra >

espandere (x+3)^{12}

Soluzione

espandere

(x + 3)^{12}

Soluzione

x^{12} + 36 x^{11} + 594 x^{10} + 5940 x^{9} + 40095 x^{8} + 192456 x^{7} + 673596 x^{6} + 1732104 x^{5} + 3247695 x^{4} + 4330260 x^{3} + 3897234 x^{2} + 2125764 x + 531441

Fasi della soluzione

(x + 3)^{12}

Applicare il teorema binomiale:

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

a = x, b = 3

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

Espandere la sommatoria

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

Semplificare

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

Semplificare

\frac{12 !}{1 ! ( 12 - 1 ) !} x^{11} \cdot 3^{1} : 36 x^{11}

Semplificare

\frac{12 !}{2 ! ( 12 - 2 ) !} x^{10} \cdot 3^{2} : 594 x^{10}

Semplificare

\frac{12 !}{3 ! ( 12 - 3 ) !} x^{9} \cdot 3^{3} : 5940 x^{9}

Semplificare

\frac{12 !}{4 ! ( 12 - 4 ) !} x^{8} \cdot 3^{4} : 40095 x^{8}

Semplificare

\frac{12 !}{5 ! ( 12 - 5 ) !} x^{7} \cdot 3^{5} : 192456 x^{7}

Semplificare

\frac{12 !}{6 ! ( 12 - 6 ) !} x^{6} \cdot 3^{6} : 673596 x^{6}

Semplificare

\frac{12 !}{7 ! ( 12 - 7 ) !} x^{5} \cdot 3^{7} : 1732104 x^{5}

Semplificare

\frac{12 !}{8 ! ( 12 - 8 ) !} x^{4} \cdot 3^{8} : 3247695 x^{4}

Semplificare

\frac{12 !}{9 ! ( 12 - 9 ) !} x^{3} \cdot 3^{9} : 4330260 x^{3}

Semplificare

\frac{12 !}{10 ! ( 12 - 10 ) !} x^{2} \cdot 3^{10} : 3897234 x^{2}

Semplificare

\frac{12 !}{11 ! ( 12 - 11 ) !} x^{1} \cdot 3^{11} : 2125764 x

Semplificare

\frac{12 !}{12 ! ( 12 - 12 ) !} x^{0} \cdot 3^{12} : 531441

= x^{12} + 36 x^{11} + 594 x^{10} + 5940 x^{9} + 40095 x^{8} + 192456 x^{7} + 673596 x^{6} + 1732104 x^{5} + 3247695 x^{4} + 4330260 x^{3} + 3897234 x^{2} + 2125764 x + 531441

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