Question Number 35419 by Rio Mike last updated on 18/May/18
$${Given}\:{that}\: \\ $$$${f}\left({x}\right)=\:{x}^{\mathrm{3}} −{x}^{\mathrm{2}} +{ax}+{b}\:{and}\: \\ $$$${g}\left({x}\right)=\:\mathrm{2}{x}^{\mathrm{3}} −\mathrm{9}{x}^{\mathrm{2}} −\mathrm{3}{ax}\:+\:{b}\:{have}\:{a} \\ $$$${common}\:{factor}\:\left({x}−\mathrm{1}\right)\:{where}\:{a}\:{and} \\ $$$${b}\:{are}\:{constands}\:.\:{Find}\:{the}\:{values} \\ $$$${of}\:{a}\:{and}\:{b}\:{hence}\:{find}\:{other}\:{factors} \\ $$$${of}\:{f}\left({x}\right) \\ $$
Answered by ajfour last updated on 18/May/18
$${a}+{b}=\mathrm{0} \\ $$$${b}−\mathrm{3}{a}=\mathrm{7} \\ $$$$\Rightarrow\:\:{a}=\frac{−\mathrm{7}}{\mathrm{4}}\:,\:\:{b}=\frac{\mathrm{7}}{\mathrm{4}} \\ $$$${f}\left({x}\right)={x}^{\mathrm{2}} \left({x}−\mathrm{1}\right)−\frac{\mathrm{7}}{\mathrm{4}}\left({x}−\mathrm{1}\right) \\ $$$$\:{f}\left({x}\right)\:\:=\left({x}+\frac{\sqrt{\mathrm{7}}}{\mathrm{2}}\right)\left({x}−\mathrm{1}\right)\left({x}−\frac{\sqrt{\mathrm{7}}}{\mathrm{2}}\right) \\ $$