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Popolare Algebra >

espandere (a^2-b^2)^6

Soluzione

espandere

(a^{2} - b^{2})^{6}

Soluzione

a^{12} - 6 a^{10} b^{2} + 15 a^{8} b^{4} - 20 a^{6} b^{6} + 15 a^{4} b^{8} - 6 a^{2} b^{10} + b^{12}

Fasi della soluzione

(a^{2} - b^{2})^{6}

Applicare il teorema binomiale:

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

a = a^{2}, b = - b^{2}

= i = 0 \sum 6 (i 6) (a^{2})^{(6 - i)} (- b^{2})^{i}

Espandere la sommatoria

= \frac{6 !}{0 ! ( 6 - 0 ) !} (a^{2})^{6} (- b^{2})^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} (a^{2})^{5} (- b^{2})^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} (a^{2})^{4} (- b^{2})^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} (a^{2})^{3} (- b^{2})^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} (a^{2})^{2} (- b^{2})^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} (a^{2})^{1} (- b^{2})^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} (a^{2})^{0} (- b^{2})^{6}

Semplificare

\frac{6 !}{0 ! ( 6 - 0 ) !} (a^{2})^{6} (- b^{2})^{0} : a^{12}

Semplificare

\frac{6 !}{1 ! ( 6 - 1 ) !} (a^{2})^{5} (- b^{2})^{1} : - 6 a^{10} b^{2}

Semplificare

\frac{6 !}{2 ! ( 6 - 2 ) !} (a^{2})^{4} (- b^{2})^{2} : 15 a^{8} b^{4}

Semplificare

\frac{6 !}{3 ! ( 6 - 3 ) !} (a^{2})^{3} (- b^{2})^{3} : - 20 a^{6} b^{6}

Semplificare

\frac{6 !}{4 ! ( 6 - 4 ) !} (a^{2})^{2} (- b^{2})^{4} : 15 a^{4} b^{8}

Semplificare

\frac{6 !}{5 ! ( 6 - 5 ) !} (a^{2})^{1} (- b^{2})^{5} : - 6 a^{2} b^{10}

Semplificare

\frac{6 !}{6 ! ( 6 - 6 ) !} (a^{2})^{0} (- b^{2})^{6} : b^{12}

= a^{12} - 6 a^{10} b^{2} + 15 a^{8} b^{4} - 20 a^{6} b^{6} + 15 a^{4} b^{8} - 6 a^{2} b^{10} + b^{12}

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