This is a tricky one

Algebra Level 4

If the equation x 4 − 4 x 3 + a x 2 + b x + 1 x^4-4x^3+ax^2+bx+1 has four positive roots, then a − b a-b is equal to

4 10 6 15

Deepansh Jindal by Deepansh Jindal , 19, India

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1 solution

Relevant wiki: Vieta's Formula Problem Solving - Intermediate

Let x 1 , x 2 , x 3 , x 4 be the roots and we have \text{Let }x_1,x_2,x_3,x_4 \text{ be the roots and we have }

{ x 1 + x 2 + x 3 + x 4 = 4 x 1 x 2 x 3 x 4 = 1 \begin{cases} x_1+x_2+x_3+x_4=4 \\ x_1x_2x_3x_4=1\end{cases}

We know by AM-GM that x 1 + x 2 + x 3 + x 4 ≥ 4 ( x 1 x 2 x 3 x 4 ) 1 4 = 4 x_1+x_2+x_3+x_4 \ge 4(x_1x_2x_3x_4)^{\frac{1}{4}}=4 & equality occurs when x 1 = x 2 = x 3 = x 4 = 1 x_1=x_2=x_3=x_4=1

Since equality occurs here i.e x 1 + x 2 + x 3 + x 4 = 4 x_1+x_2+x_3+x_4=4 we have four equal roots .

x 4 − 4 x 3 + a x 2 + b x + 1 = ( x − 1 ) ( x − 1 ) ( x − 1 ) ( x − 1 ) = x 4 − 4 x 3 + 6 x 2 − 4 x + 1 x^4-4x^3+ax^2+bx+1 = (x-1)(x-1)(x-1)(x-1) = x^4-4x^3+6x^2-4x+1 , Comparing coefficients we get a = 6 \boxed{a=6} & b = − 4 \boxed{b=-4} . Thus a − b = 10 \boxed{a-b=10}

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How you got inspired to use AM - GM inequality \text{AM - GM inequality} ??

Chirayu Bhardwaj - 5 years, 1 month ago

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Actually, whenever in a problem we have relations between sum & product of a finite number of variables, inequalities come handy in these cases, the most classical case being these where it holds too. @Chirayu Bhardwaj

Aditya Narayan Sharma - 5 years, 1 month ago

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