Uwolnić od niewymierności ułamek w; a: P(a)=0

[mol_ksiazkowy]

\(\displaystyle{ P(x)=x^{3}-3x^{2}+x+1}\)
\(\displaystyle{ w= \frac{1}{2+3a-a^{2}}}\)

[jasny]

\(\displaystyle{ P(a)=a^3-3a^2+a+1=(a-1)(a^2-2a-1)=(a-1)(a-(1-\sqrt2))(a-(1+\sqrt2))}\)
\(\displaystyle{ P(a)=0\:\Leftrightarrow\:a=1\,\vee\,a=1-\sqrt2\,\vee\,a=1+\sqrt2}\)
\(\displaystyle{ w=\frac{1}{2+3a-a^2}}\)
\(\displaystyle{ w=\frac{1}{4}\,\vee\,w=\frac{1}{2+3-3\sqrt2-1+2\sqrt2-2}\,\vee\,w=\frac{1}{2+3+3\sqrt2-1-2\sqrt2-2}}\)
\(\displaystyle{ w=\frac{1}{4}\,\vee\,w=\frac{1}{2-\sqrt2}\,\vee\,w=\frac{1}{2+\sqrt2}}\)
\(\displaystyle{ w=\frac{1}{4}\,\vee\,w=\frac{2+\sqrt2}{2}\,\vee\,w=\frac{2-\sqrt2}{2}}\)