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suppose that aii roots of the polynomial equation x^4 - 4x^3 + ax^2 + bx + 1=0 are positive real numbers. show that all the roots of the polynomial are equal.

suppose that aii roots of the polynomial equation


x^4 - 4x^3 + ax^2 + bx + 1=0  are positive real numbers. show that all the roots of the polynomial are equal.

Grade:11

1 Answers

Swapnil Saxena
102 Points
9 years ago

This can be solved by AM-GM inequality

Let the roots be a1,a2,a3,a4

Then a1*a2*a3*a4 = 1/1

Also (-1)(a1+a2+a3+a4) = -4/1= 4

If a1,a2,a3,a4 are all positive roots of the equation, applying AM-GM inequality

now if (a1+a2+a3+a4)/4= (a1*a2*a3*a4)^1/4 then essentially a1=a2=a3=a4

Then all rotts r equal

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