The roots of the equation $x^4-8x^3+ax^2-bx+16=0$ are positive if
(A) a=24
(B) a=12
(C) b=8
(D) b=32

The hint given in the book says
"Roots are $\alpha_1,\alpha_2,\alpha_3,\alpha_4,$
AM=GM=2, hence all the roots are equal."

Now I don't know what to infer from the hint. Can someone help me solve this question

  • $\begingroup$ Arithmetic and geometric means are equal and 2: $\sum_{i=1}^{4} {x_{i}} = ½$, $\prod_{i=1}^{4} {x_{i}} = 16$. But what's the question? Whether the claim is true? $\endgroup$ – Dohleman Mar 11 '17 at 7:15

Note that (by using Vieta's Formula)

$$\alpha_1+\alpha_2+\alpha_3+\alpha_4=8$$ and $$\alpha_1\alpha_2\alpha_3\alpha_4=16$$

Therefore $AM=GM=2$

Deducing from $AM-GM$ inequality $\left( \frac{\sum_{i=1}^na_{i}}{n}\geq (\prod_{i=1}^na_i)^{1/n}\right)$, $AM=GM$ only when all the terms are equal..Hence all roots are equal.

So $\alpha_1=2$ And $x^4-8x^3+ax^2-bx+16=(x-2)^4$

Multiply and compare coefficients.

So, $(A)$ and $(D)$ are true...

Hope that helps.


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