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Populaire Algèbre >

développer (1+x+1/2 x^2)^8

Solution

développer

(1 + x + \frac{1}{2} x^{2})^{8}

Solution

\frac{x ^{16}}{256} + \frac{x ^{15}}{16} + \frac{x ^{14}}{2} + \frac{21 x ^{13}}{8} + \frac{161 x ^{12}}{16} + \frac{119 x ^{11}}{4} + 70 x^{10} + \frac{267 x ^{9}}{2} + \frac{1667 x ^{8}}{8} + 267 x^{7} + 280 x^{6} + 238 x^{5} + 161 x^{4} + 84 x^{3} + 32 x^{2} + 8 x + 1

étapes des solutions

(1 + x + \frac{1}{2} x^{2})^{8}

\frac{1}{2} x^{2} = \frac{x ^{2}}{2}

= (1 + x + \frac{x ^{2}}{2})^{8}

(1 + x + \frac{x ^{2}}{2})^{8} = (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2})

= (\frac{x ^{2}}{2} + x + 1) (\frac{x ^{2}}{2} + x + 1) (\frac{x ^{2}}{2} + x + 1) (\frac{x ^{2}}{2} + x + 1) (\frac{x ^{2}}{2} + x + 1) (\frac{x ^{2}}{2} + x + 1) (\frac{x ^{2}}{2} + x + 1) (\frac{x ^{2}}{2} + x + 1)

Développer

(1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) (1 + x + \frac{x ^{2}}{2}) : \frac{x ^{16} + 16 x ^{15} + 128 x ^{14} + 672 x ^{13} + 2576 x ^{12} + 7616 x ^{11} + 17920 x ^{10} + 34176 x ^{9} + 53344 x ^{8} + 68352 x ^{7} + 71680 x ^{6} + 60928 x ^{5} + 41216 x ^{4} + 21504 x ^{3} + 8192 x ^{2} + 2048 x}{256} + 1

= \frac{x ^{16} + 16 x ^{15} + 128 x ^{14} + 672 x ^{13} + 2576 x ^{12} + 7616 x ^{11} + 17920 x ^{10} + 34176 x ^{9} + 53344 x ^{8} + 68352 x ^{7} + 71680 x ^{6} + 60928 x ^{5} + 41216 x ^{4} + 21504 x ^{3} + 8192 x ^{2} + 2048 x}{256} + 1

Appliquer la règle des fractions:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{x ^{16} + 16 x ^{15} + 128 x ^{14} + 672 x ^{13} + 2576 x ^{12} + 7616 x ^{11} + 17920 x ^{10} + 34176 x ^{9} + 53344 x ^{8} + 68352 x ^{7} + 71680 x ^{6} + 60928 x ^{5} + 41216 x ^{4} + 21504 x ^{3} + 8192 x ^{2} + 2048 x}{256} = \frac{x ^{16}}{256} + \frac{16 x ^{15}}{256} + \frac{128 x ^{14}}{256} + \frac{672 x ^{13}}{256} + \frac{2576 x ^{12}}{256} + \frac{7616 x ^{11}}{256} + \frac{17920 x ^{10}}{256} + \frac{34176 x ^{9}}{256} + \frac{53344 x ^{8}}{256} + \frac{68352 x ^{7}}{256} + \frac{71680 x ^{6}}{256} + \frac{60928 x ^{5}}{256} + \frac{41216 x ^{4}}{256} + \frac{21504 x ^{3}}{256} + \frac{8192 x ^{2}}{256} + \frac{2048 x}{256}

= \frac{x ^{16}}{256} + \frac{16 x ^{15}}{256} + \frac{128 x ^{14}}{256} + \frac{672 x ^{13}}{256} + \frac{2576 x ^{12}}{256} + \frac{7616 x ^{11}}{256} + \frac{17920 x ^{10}}{256} + \frac{34176 x ^{9}}{256} + \frac{53344 x ^{8}}{256} + \frac{68352 x ^{7}}{256} + \frac{71680 x ^{6}}{256} + \frac{60928 x ^{5}}{256} + \frac{41216 x ^{4}}{256} + \frac{21504 x ^{3}}{256} + \frac{8192 x ^{2}}{256} + \frac{2048 x}{256} + 1

\frac{16 x ^{15}}{256} = \frac{x ^{15}}{16}

\frac{128 x ^{14}}{256} = \frac{x ^{14}}{2}

\frac{672 x ^{13}}{256} = \frac{21 x ^{13}}{8}

\frac{2576 x ^{12}}{256} = \frac{161 x ^{12}}{16}

\frac{7616 x ^{11}}{256} = \frac{119 x ^{11}}{4}

\frac{17920 x ^{10}}{256} = 70 x^{10}

\frac{34176 x ^{9}}{256} = \frac{267 x ^{9}}{2}

\frac{53344 x ^{8}}{256} = \frac{1667 x ^{8}}{8}

\frac{68352 x ^{7}}{256} = 267 x^{7}

\frac{71680 x ^{6}}{256} = 280 x^{6}

\frac{60928 x ^{5}}{256} = 238 x^{5}

\frac{41216 x ^{4}}{256} = 161 x^{4}

\frac{21504 x ^{3}}{256} = 84 x^{3}

\frac{8192 x ^{2}}{256} = 32 x^{2}

\frac{2048 x}{256} = 8 x

= \frac{x ^{16}}{256} + \frac{x ^{15}}{16} + \frac{x ^{14}}{2} + \frac{21 x ^{13}}{8} + \frac{161 x ^{12}}{16} + \frac{119 x ^{11}}{4} + 70 x^{10} + \frac{267 x ^{9}}{2} + \frac{1667 x ^{8}}{8} + 267 x^{7} + 280 x^{6} + 238 x^{5} + 161 x^{4} + 84 x^{3} + 32 x^{2} + 8 x + 1

Exemples populaires

développer-(4(3x^2-19x+75))/((x+5)(x-2)(x-9))

e x p an d - \frac{4 ( 3 x ^{2} - 19 x + 75 )}{( x + 5 ) ( x - 2 ) ( x - 9 )}

(\partial)/(\partial x)((z)(5x+8y)^{11})

\frac{\partial}{\partial x} ((z) (5 x + 8 y)^{11})

développer (6x^2-2)/(x^2(x+1)^2(x-1)^2)

e x p an d \frac{6 x ^{2} - 2}{x ^{2} ( x + 1 ) ^{2} ( x - 1 ) ^{2}}

développer 5/((x^2+1)(x^2+16))

e x p an d \frac{5}{( x ^{2} + 1 ) ( x ^{2} + 16 )}

développer 1-(a+bx)(c+dx)

e x p an d 1 - (a + b x) (c + d x)