Dam Naj, Pilne Wykaż że jeżeli a-b=x, a^2-b^2=y i a^3-b^3=z to z=(x^4+3y^2)/4x

[latex]a-b=x /()^2[/latex] [latex]a^2-2ab+b^2=x^2[/latex] [latex]a^2+b^2=x^2+2ab[/latex] ----------------- [latex]a^2-b^2=y[/latex] [latex](a-b)(a+b)=y[/latex] [latex]x(a+b)=y /:x[/latex] [latex]a+b= frac{y}{x} /()^2[/latex] [latex]a^2+2ab+b^2= frac{y^2}{x^2}[/latex] [latex](a^2+b^2)+2ab= frac{y^2}{x^2}[/latex] Obliczam [latex]ab[/latex] podstawiamy [latex]a^2+b^2=x^2+2ab[/latex] [latex]x^2+2ab+2ab= frac{y^2}{x^2}[/latex] [latex]4ab= frac{y^2}{x^2}-x^2[/latex] [latex]4ab= frac{y^2-x^4}{x^2} /:4[/latex] [latex]ab=frac{y^2-x^4}{4x^2}[/latex] Obliczam [latex]a^2+b^2[/latex] [latex]a^2+b^2=x^2+2ab[/latex] [latex]a^2+b^2=x^2+2 cdot frac{y^2-x^4}{4x^2}[/latex] [latex]a^2+b^2=x^2+frac{y^2-x^4}{2x^2}[/latex] [latex]a^2+b^2=frac{2x^4+y^2-x^4}{2x^2}[/latex] [latex]a^2+b^2=frac{x^4+y^2}{2x^2}[/latex] ----------------- Obliczam [latex]z[/latex] [latex]a^3-b^3=z[/latex] [latex](a - b)(a^2 + ab + b^2)=z[/latex] [latex]x(a^2 + ab + b^2)=z[/latex] [latex]xleft(frac{x^4+y^2}{2x^2}+frac{y^2-x^4}{4x^2} ight) =z[/latex] [latex]z=xleft(frac{2(x^4+y^2)}{4x^2}+frac{y^2-x^4}{4x^2} ight)[/latex] [latex]z=xleft(frac{2x^4+2y^2+y^2-x^4}{4x^2} ight)[/latex] [latex]z=xleft(frac{x^4+3y^2}{4x^2} ight)[/latex] [latex]z= frac{x^4+3y^2}{4x}[/latex]