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Populaire Algèbre >

développer (w^2+1)^8

Solution

développer

(w^{2} + 1)^{8}

Solution

w^{16} + 8 w^{14} + 28 w^{12} + 56 w^{10} + 70 w^{8} + 56 w^{6} + 28 w^{4} + 8 w^{2} + 1

étapes des solutions

(w^{2} + 1)^{8}

Appliquer le théorème du binôme:

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

a = w^{2}, b = 1

= i = 0 \sum 8 (i 8) (w^{2})^{(8 - i)} \cdot 1^{i}

Développer la somme

= \frac{8 !}{0 ! ( 8 - 0 ) !} (w^{2})^{8} \cdot 1^{0} + \frac{8 !}{1 ! ( 8 - 1 ) !} (w^{2})^{7} \cdot 1^{1} + \frac{8 !}{2 ! ( 8 - 2 ) !} (w^{2})^{6} \cdot 1^{2} + \frac{8 !}{3 ! ( 8 - 3 ) !} (w^{2})^{5} \cdot 1^{3} + \frac{8 !}{4 ! ( 8 - 4 ) !} (w^{2})^{4} \cdot 1^{4} + \frac{8 !}{5 ! ( 8 - 5 ) !} (w^{2})^{3} \cdot 1^{5} + \frac{8 !}{6 ! ( 8 - 6 ) !} (w^{2})^{2} \cdot 1^{6} + \frac{8 !}{7 ! ( 8 - 7 ) !} (w^{2})^{1} \cdot 1^{7} + \frac{8 !}{8 ! ( 8 - 8 ) !} (w^{2})^{0} \cdot 1^{8}

Simplifier

\frac{8 !}{0 ! ( 8 - 0 ) !} (w^{2})^{8} \cdot 1^{0} : w^{16}

Simplifier

\frac{8 !}{1 ! ( 8 - 1 ) !} (w^{2})^{7} \cdot 1^{1} : 8 w^{14}

Simplifier

\frac{8 !}{2 ! ( 8 - 2 ) !} (w^{2})^{6} \cdot 1^{2} : 28 w^{12}

Simplifier

\frac{8 !}{3 ! ( 8 - 3 ) !} (w^{2})^{5} \cdot 1^{3} : 56 w^{10}

Simplifier

\frac{8 !}{4 ! ( 8 - 4 ) !} (w^{2})^{4} \cdot 1^{4} : 70 w^{8}

Simplifier

\frac{8 !}{5 ! ( 8 - 5 ) !} (w^{2})^{3} \cdot 1^{5} : 56 w^{6}

Simplifier

\frac{8 !}{6 ! ( 8 - 6 ) !} (w^{2})^{2} \cdot 1^{6} : 28 w^{4}

Simplifier

\frac{8 !}{7 ! ( 8 - 7 ) !} (w^{2})^{1} \cdot 1^{7} : 8 w^{2}

Simplifier

\frac{8 !}{8 ! ( 8 - 8 ) !} (w^{2})^{0} \cdot 1^{8} : 1

= w^{16} + 8 w^{14} + 28 w^{12} + 56 w^{10} + 70 w^{8} + 56 w^{6} + 28 w^{4} + 8 w^{2} + 1

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