Question

Solve the equations:$$x^{4} - 16x^{3} + 86x^{2} - 176x + 105 = 0$$, two roots being $$1$$ and $$7$$.

Solution

$${ x }^{ 4 }-16{ x }^{ 3 }+86{ x }^{ 2 }-176x+105=0$$Let the other two roots be $$a,b$$$${ S }_{ 1 }=1+7+a+b=16\\ \Rightarrow a+b=8....(i)\\ { S }_{ 4 }=7ab=105\\ \Rightarrow ab=15.....(ii)$$substituting $$b$$ from $$(i)$$$$a(8-a)=15\\ 8a-{ a }^{ 2 }=15\\ { a }^{ 2 }-8a+15=0\\ { a }^{ 2 }-5a-3a+15=0\\ a(a-5)-3(a-5)=0\\ (a-3)(a-5)=0\\ \Rightarrow a=3,5$$substituting $$a$$ in $$(ii)$$$$\Rightarrow b=5,3$$So the other two roots are $$3$$ and $$5$$Mathematics

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