expand ((1+x^2)^7)/(x^2)

פתרון

expand

\frac{( 1 + x ^{2} ) ^{7}}{x ^{2}}

\frac{1}{x ^{2}} + 7 + 21 x^{2} + 35 x^{4} + 35 x^{6} + 21 x^{8} + 7 x^{10} + x^{12}

צעדי פתרון

\frac{( 1 + x ^{2} ) ^{7}}{x ^{2}}

(1 + x^{2})^{7} = 1 + 7 x^{2} + 21 x^{4} + 35 x^{6} + 35 x^{8} + 21 x^{10} + 7 x^{12} + x^{14}

= \frac{1 + 7 x ^{2} + 21 x ^{4} + 35 x ^{6} + 35 x ^{8} + 21 x ^{10} + 7 x ^{12} + x ^{14}}{x ^{2}}

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

: השתמש בתכונת השברים הבאה

\frac{1 + 7 x ^{2} + 21 x ^{4} + 35 x ^{6} + 35 x ^{8} + 21 x ^{10} + 7 x ^{12} + x ^{14}}{x ^{2}} = \frac{1}{x ^{2}} + \frac{7 x ^{2}}{x ^{2}} + \frac{21 x ^{4}}{x ^{2}} + \frac{35 x ^{6}}{x ^{2}} + \frac{35 x ^{8}}{x ^{2}} + \frac{21 x ^{10}}{x ^{2}} + \frac{7 x ^{12}}{x ^{2}} + \frac{x ^{14}}{x ^{2}}

= \frac{1}{x ^{2}} + \frac{7 x ^{2}}{x ^{2}} + \frac{21 x ^{4}}{x ^{2}} + \frac{35 x ^{6}}{x ^{2}} + \frac{35 x ^{8}}{x ^{2}} + \frac{21 x ^{10}}{x ^{2}} + \frac{7 x ^{12}}{x ^{2}} + \frac{x ^{14}}{x ^{2}}

\frac{7 x ^{2}}{x ^{2}} = 7

\frac{21 x ^{4}}{x ^{2}} = 21 x^{2}

\frac{35 x ^{6}}{x ^{2}} = 35 x^{4}

\frac{35 x ^{8}}{x ^{2}} = 35 x^{6}

\frac{21 x ^{10}}{x ^{2}} = 21 x^{8}

\frac{7 x ^{12}}{x ^{2}} = 7 x^{10}

\frac{x ^{14}}{x ^{2}} = x^{12}

= \frac{1}{x ^{2}} + 7 + 21 x^{2} + 35 x^{4} + 35 x^{6} + 21 x^{8} + 7 x^{10} + x^{12}

s im pl i f y (x + \frac{1}{2}) (x - \frac{1}{3})

e x p an d \frac{x ^{3} + 2 x - 1}{x ( x - 1 ) ^{2} ( x + 2 )}

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

s im pl i f y (x + 8) (x - \frac{2}{5}) (x + 2) (x - 2)

e x p an d \frac{x + h - x}{h}