# AB: Gemischte Rechenaufgaben XIII

## I. Polynomdivision

1.

Führen Sie die Polynomdivision aus. Beachten Sie, dass auch ein Rest bleiben kann.

a)

$$\frac{4 x^{2}-4}{x-1}$$

$$\left(4 x^{2}-4\right):(x-1) = 4x + 4 \\ \frac{- \left( 4 x^{2}-4 x\right)}{ \qquad 4 x-4} \\ \frac{ \quad -(4 x-4)}{ \qquad \quad 0 }$$

b)

$$\frac{3 x^{2}+6 x}{2 x^{2}-1}$$

$$\left(3x^{2} + 6x\right):(2x^2-1) = \frac{3}{2} + \frac{6x + \frac{3}{2}}{2x^2 - 1} \\ \frac{- \left( 3 x^{2} - \frac{3}{2} \right)}{ \qquad 6x + \frac{3}{2} } _{ \quad ← \text{ Rest } }$$

c)

$$\frac{8 x+4}{6 x}$$

$$\left(8x + 4\right): 6x = \frac{8}{6} + \frac{4}{6x} = \frac{4}{3} + \frac{2}{3x} \\ \frac{-(8x) }{ \qquad 4 }_{ \quad ← \text{ Rest } }$$

d)

$$\frac{3 x^{2}+5 x-3}{4 x^{2}-2}$$

$$\left(3x^{2} + 5x - 3\right):(4x^2-2) = \frac{3}{4} + \frac{5x - \frac{3}{2}}{4x^2 - 2} \\ \frac{- \left( 3 x^{2} - \frac{3}{2} \right)}{ \qquad 5x - \frac{3}{2} } _{ \quad ← \text{ Rest } }$$

e)

$$\frac{x^{3}-5 x^{2}+4 x-3}{x+1}$$

$$\left(x-5 x^{2}+4 x-3\right):(x+1) = x^2 - 6x + 10 - \frac{13}{x+1} \\ \frac{-\left(x^{3}+x\right)}{-6 x^{2}+4 x} \\ \quad \frac{ -\left(-6 x^{2}-6 x\right)}{ \qquad \qquad 10 x-3} \\ \qquad \qquad \frac{-10 x+10}{ \qquad -13}$$

f)

$$\frac{x^{2}-4}{x+2}$$

$$(x^2 - 4):(x+2) = x - 2 \\ \frac{-(x^{2}+2 x_{1})}{ \qquad -2 x-4} \\ \quad \frac{-(-2 x-4)}{ \qquad 0}$$

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