Wiemy, że:

$w\left(x\right)=x^3-1$

$p\left(x\right)=2x^2+4x+1$

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a)

Wyznaczamy wielomian $u$:

$u\left(x\right)=2w\left(x\right)+\left(1-x\right)p\left(x\right)=2\left(x^3-1\right)+\left(1-x\right)\left(2x^2+4x+1\right)=$

$=2x^3-2+2x^2+4x+1-2x^3-4x^2-x=$

$=-2x^2+3x-1$

Odczytujemy:

$\text{st}\left(u\right)=2$

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b)

Wyznaczamy wielomian $u$:

$u\left(x\right)={\left[w\left(x\right)\right]}^2-\frac{x^4}{2}\cdot p\left(x\right)={\left(x^3-1\right)}^2-\frac{x^4}{2}\left(2x^2+4x+1\right)=$

$=x^6-2x^3+1-x^6-2x^5-\frac{1}{2}{x}^4=-2x^5-\frac{1}{2}{x}^4-2x^3+1$

Odczytujemy:

$\text{st}\left(u\right)=5$