Dane są wielomiany [latex]W(x)=(x+1)(x+2)(x+3) [/latex] oraz [latex]Z(x)= x^{3} +(m+k) x^{2} +(2k-5m)x+6[/latex] a) wyznacz tak liczby k i m, aby te wielomiany były równe b) oblicz [latex]W( sqrt{2} -2)[/latex]

[latex]W(x)=(x+1)(x+2)(x+3)=(x^2+3x+2)(x+3)=\=x^3+3x^2+3x^2+9x+2x+6=x^3+6x^2+11x+6\ \ Z(x)= x^{3} +(m+k) x^{2} +(2k-5m)x+6\ \ egin{cases}m+k=6 \2k-5m=11end{cases}\ egin{cases}m=6-k\2k-5(6-k)=11end{cases}\ 2k-30+5k=11\ 7k=41\ k=frac{41}{7}=5frac{6}{7}\ \ m=6-5frac{6}{7}=frac{1}{7}\ \ egin{cases}m=frac{1}{7}\k=5frac{6}{7}end{cases} [/latex] [latex]W(sqrt{2}-2)=x^3+6x^2+11x+6=(sqrt{2}-2)^3+6(sqrt{2}-2)^2+\+11(sqrt{2}-2)+6=2sqrt{2}-12+12sqrt{2}-8+12-24sqrt{2}+24+\+11sqrt{2}-22+6=sqrt{2}[/latex]