a)

Dany jest wielomian $w$ określony wzorem

$w\left(x,y\right)=x^3{y}^2-3x{y}^4+4xy$

Obliczamy wartość wielomianu dla $x=2$ i $y=-\frac{1}{2}$.

$w\left(2,-\frac{1}{2}\right)=2^3\cdot {\left(-\frac{1}{2}\right)}^2-3\cdot 2\cdot {\left(-\frac{1}{2}\right)}^4+4\cdot 2\cdot \left(-\frac{1}{2}\right)=$

$=8\cdot \frac{1}{4}-3\cdot 2\cdot \frac{1}{16}-4=2-\frac{3}{8}-4=-2\frac{3}{8}$

Obliczamy wartość wielomianu dla $x=-\sqrt{2}$ i $y=\sqrt{6}$.

$w\left(-\sqrt{2},\sqrt{6}\right)=$

$={\left(-\sqrt{2}\right)}^3\cdot {\left(\sqrt{6}\right)}^2-3\cdot \left(-\sqrt{2}\right)\cdot {\left(-\sqrt{6}\right)}^4+4\cdot \left(-\sqrt{2}\right)\cdot \sqrt{6}=$

$=-2\sqrt{2}\cdot 6+3\cdot \sqrt{2}\cdot 36-4\sqrt{12}=-12\sqrt{2}+108\sqrt{2}-8\sqrt{3}=$

$=98\sqrt{2}-8\sqrt{3}$

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b)

Dany jest wielomian

$w\left(x,y\right)=\frac{1}{2}{x}^3{y}^3-2x^{2}y+4x{y}^3$

Obliczamy wartość wielomianu dla $x=2$ i $y=-\frac{1}{2}$.

$w\left(2,-\frac{1}{2}\right)=\frac{1}{2}\cdot 2^3\cdot {\left(-\frac{1}{2}\right)}^3-2\cdot 2^2\cdot \left(-\frac{1}{2}\right)+4\cdot 2\cdot {\left(-\frac{1}{2}\right)}^3=$

$=\frac{1}{2}\cdot 8\cdot \left(-\frac{1}{8}\right)+2\cdot 4\cdot \frac{1}{2}+4\cdot 2\cdot \left(-\frac{1}{8}\right)=-\frac{1}{2}+4-1=2\frac{1}{2}$

Obliczamy wartość wielomianu dla $x=-\sqrt{2}$ i $y=\sqrt{6}$.

$w\left(-\sqrt{2},\sqrt{6}\right)=$

$=\frac{1}{2}\cdot {\left(-\sqrt{2}\right)}^3\cdot {\left(\sqrt{6}\right)}^3-2\cdot {\left(-\sqrt{2}\right)}^2\cdot \sqrt{6}+4\cdot \left(-\sqrt{2}\right)\cdot {\left(\sqrt{6}\right)}^3=$

$=\frac{1}{2}\cdot \left(-2\sqrt{2}\right)\cdot 6\sqrt{6}-2\cdot 2\cdot \sqrt{6}+4\cdot \left(-\sqrt{2}\right)\cdot 6\sqrt{6}=$

$=-6\sqrt{12}-4\sqrt{6}-24\sqrt{12}=-30\sqrt{4\cdot 3}-4\sqrt{6}=-60\sqrt{3}-4\sqrt{6}$