erweitern ((x+h)^3-3ax^2+2bx+c)/h

Lösung

erweitern

\frac{( x + h ) ^{3} - 3 a x ^{2} + 2 b x + c}{h}

\frac{x ^{3}}{h} + 3 x^{2} + 3 x h + h^{2} - \frac{3 a x ^{2}}{h} + \frac{2 b x}{h} + \frac{c}{h}

Schritte zur Lösung

\frac{( x + h ) ^{3} - 3 a x ^{2} + 2 b x + c}{h}

Multipliziere aus

(x + h)^{3} - 3 a x^{2} + 2 b x + c : x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 3 a x^{2} + 2 b x + c

= \frac{x ^{3} + 3 x ^{2} h + 3 x h ^{2} + h ^{3} - 3 a x ^{2} + 2 b x + c}{h}

Wende Bruchregel an:

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

\frac{x ^{3} + 3 x ^{2} h + 3 x h ^{2} + h ^{3} - 3 a x ^{2} + 2 b x + c}{h} = \frac{x ^{3}}{h} + \frac{3 x ^{2} h}{h} + \frac{3 x h ^{2}}{h} + \frac{h ^{3}}{h} - \frac{3 a x ^{2}}{h} + \frac{2 b x}{h} + \frac{c}{h}

= \frac{x ^{3}}{h} + \frac{3 x ^{2} h}{h} + \frac{3 x h ^{2}}{h} + \frac{h ^{3}}{h} - \frac{3 a x ^{2}}{h} + \frac{2 b x}{h} + \frac{c}{h}

Streiche

\frac{3 x ^{2} h}{h} : 3 x^{2}

= \frac{x ^{3}}{h} + 3 x^{2} + \frac{3 x h ^{2}}{h} + \frac{h ^{3}}{h} - \frac{3 a x ^{2}}{h} + \frac{2 b x}{h} + \frac{c}{h}

Streiche

\frac{3 x h ^{2}}{h} : 3 x h

= \frac{x ^{3}}{h} + 3 x^{2} + 3 x h + \frac{h ^{3}}{h} - \frac{3 a x ^{2}}{h} + \frac{2 b x}{h} + \frac{c}{h}

Streiche

\frac{h ^{3}}{h} : h^{2}

= \frac{x ^{3}}{h} + 3 x^{2} + 3 x h + h^{2} - \frac{3 a x ^{2}}{h} + \frac{2 b x}{h} + \frac{c}{h}

e x p an d \frac{3 x ^{2} + 5}{2 ( x - 1 ) ^{2}}

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

e x p an d (3 - 11 - x) (6 - x + 2)

e x p an d (e^{- t} + 2 e^{2 t}) (- 2 e^{- t} - 8 e^{- 2 t})

e x p an d \frac{4}{( x + 2 ) ( x - 2 )}