91Ó°ÊÓ

Find the equations with integral coefficients having only the following roots : (a) \(5,1,12\) (b) \((1 / 2),(1 / 3),(-1 / 4)\)

Answer the following questions about rational and real coefficient equations : (1) A real coefficient equation has 2 roots, one root is \(3-\mathrm{i} \sqrt{6}\), what is the second root? (2) A real coefficient equation has 2 roots, one root is \(-\sqrt{13}\), what is the second root? (3) Are the following statements correct and why? (a) \(x^{3}+8 x-7 i=0\) has \(x=a-b i\) as one root, therefore, the second root must be \(\mathrm{x}=\mathrm{a}+\mathrm{bi}\). (b) \(x^{3}+\sqrt{2} x^{2}+6 x+\sqrt{1} 3=0\) has one root equal to \(\mathrm{x}=\mathrm{a}+\mathrm{bi}\) therefore, the second root \(\mathrm{x}=\mathrm{a}-\mathrm{bi} .\)