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Populaire Algèbre >

développer (x^2+2sqrt(x))^{15}

Solution

développer

(x^{2} + 2 x)^{15}

Solution

x^{30} + 30 x^{57} + 420 x^{27} + 3640 x^{51} + 21840 x^{24} + 96096 x^{45} + 320320 x^{21} + 823680 x^{39} + 1647360 x^{18} + 2562560 x^{33} + 3075072 x^{15} + 2795520 x^{27} + 1863680 x^{12} + 860160 x^{21} + 245760 x^{9} + 32768 x^{15}

étapes des solutions

(x^{2} + 2 x)^{15}

Appliquer le théorème du binôme:

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

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

= i = 0 \sum 15 (i 15) (x^{2})^{(15 - i)} (2 x)^{i}

Développer la somme

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

Simplifier

\frac{15 !}{0 ! ( 15 - 0 ) !} (x^{2})^{15} (2 x)^{0} : x^{30}

Simplifier

\frac{15 !}{1 ! ( 15 - 1 ) !} (x^{2})^{14} (2 x)^{1} : 30 x^{28} x

Simplifier

\frac{15 !}{2 ! ( 15 - 2 ) !} (x^{2})^{13} (2 x)^{2} : 420 x^{27}

Simplifier

\frac{15 !}{3 ! ( 15 - 3 ) !} (x^{2})^{12} (2 x)^{3} : 3640 x^{24} (x)^{3}

Simplifier

\frac{15 !}{4 ! ( 15 - 4 ) !} (x^{2})^{11} (2 x)^{4} : 21840 x^{24}

Simplifier

\frac{15 !}{5 ! ( 15 - 5 ) !} (x^{2})^{10} (2 x)^{5} : 96096 x^{20} (x)^{5}

Simplifier

\frac{15 !}{6 ! ( 15 - 6 ) !} (x^{2})^{9} (2 x)^{6} : 320320 x^{21}

Simplifier

\frac{15 !}{7 ! ( 15 - 7 ) !} (x^{2})^{8} (2 x)^{7} : 823680 x^{16} (x)^{7}

Simplifier

\frac{15 !}{8 ! ( 15 - 8 ) !} (x^{2})^{7} (2 x)^{8} : 1647360 x^{18}

Simplifier

\frac{15 !}{9 ! ( 15 - 9 ) !} (x^{2})^{6} (2 x)^{9} : 2562560 x^{12} (x)^{9}

Simplifier

\frac{15 !}{10 ! ( 15 - 10 ) !} (x^{2})^{5} (2 x)^{10} : 3075072 x^{15}

Simplifier

\frac{15 !}{11 ! ( 15 - 11 ) !} (x^{2})^{4} (2 x)^{11} : 2795520 x^{8} (x)^{11}

Simplifier

\frac{15 !}{12 ! ( 15 - 12 ) !} (x^{2})^{3} (2 x)^{12} : 1863680 x^{12}

Simplifier

\frac{15 !}{13 ! ( 15 - 13 ) !} (x^{2})^{2} (2 x)^{13} : 860160 x^{4} (x)^{13}

Simplifier

\frac{15 !}{14 ! ( 15 - 14 ) !} (x^{2})^{1} (2 x)^{14} : 245760 x^{9}

Simplifier

\frac{15 !}{15 ! ( 15 - 15 ) !} (x^{2})^{0} (2 x)^{15} : 32768 (x)^{15}

= x^{30} + 30 x^{28} x + 420 x^{27} + 3640 x^{24} (x)^{3} + 21840 x^{24} + 96096 x^{20} (x)^{5} + 320320 x^{21} + 823680 x^{16} (x)^{7} + 1647360 x^{18} + 2562560 x^{12} (x)^{9} + 3075072 x^{15} + 2795520 x^{8} (x)^{11} + 1863680 x^{12} + 860160 x^{4} (x)^{13} + 245760 x^{9} + 32768 (x)^{15}

Simplifier

x^{30} + 30 x^{28} x + 420 x^{27} + 3640 x^{24} (x)^{3} + 21840 x^{24} + 96096 x^{20} (x)^{5} + 320320 x^{21} + 823680 x^{16} (x)^{7} + 1647360 x^{18} + 2562560 x^{12} (x)^{9} + 3075072 x^{15} + 2795520 x^{8} (x)^{11} + 1863680 x^{12} + 860160 x^{4} (x)^{13} + 245760 x^{9} + 32768 (x)^{15} : x^{30} + 30 x^{\frac{57}{2}} + 420 x^{27} + 3640 x^{\frac{51}{2}} + 21840 x^{24} + 96096 x^{\frac{45}{2}} + 320320 x^{21} + 823680 x^{\frac{39}{2}} + 1647360 x^{18} + 2562560 x^{\frac{33}{2}} + 3075072 x^{15} + 2795520 x^{\frac{27}{2}} + 1863680 x^{12} + 860160 x^{\frac{21}{2}} + 245760 x^{9} + 32768 x^{\frac{15}{2}}

= x^{30} + 30 x^{\frac{57}{2}} + 420 x^{27} + 3640 x^{\frac{51}{2}} + 21840 x^{24} + 96096 x^{\frac{45}{2}} + 320320 x^{21} + 823680 x^{\frac{39}{2}} + 1647360 x^{18} + 2562560 x^{\frac{33}{2}} + 3075072 x^{15} + 2795520 x^{\frac{27}{2}} + 1863680 x^{12} + 860160 x^{\frac{21}{2}} + 245760 x^{9} + 32768 x^{\frac{15}{2}}

Simplifier

x^{30} + 30 x^{\frac{57}{2}} + 420 x^{27} + 3640 x^{\frac{51}{2}} + 21840 x^{24} + 96096 x^{\frac{45}{2}} + 320320 x^{21} + 823680 x^{\frac{39}{2}} + 1647360 x^{18} + 2562560 x^{\frac{33}{2}} + 3075072 x^{15} + 2795520 x^{\frac{27}{2}} + 1863680 x^{12} + 860160 x^{\frac{21}{2}} + 245760 x^{9} + 32768 x^{\frac{15}{2}} : x^{30} + 30 x^{57} + 420 x^{27} + 3640 x^{51} + 21840 x^{24} + 96096 x^{45} + 320320 x^{21} + 823680 x^{39} + 1647360 x^{18} + 2562560 x^{33} + 3075072 x^{15} + 2795520 x^{27} + 1863680 x^{12} + 860160 x^{21} + 245760 x^{9} + 32768 x^{15}

= x^{30} + 30 x^{57} + 420 x^{27} + 3640 x^{51} + 21840 x^{24} + 96096 x^{45} + 320320 x^{21} + 823680 x^{39} + 1647360 x^{18} + 2562560 x^{33} + 3075072 x^{15} + 2795520 x^{27} + 1863680 x^{12} + 860160 x^{21} + 245760 x^{9} + 32768 x^{15}

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