# Consider the polynomial #f(x)=x^4-4ax^3+6b^2x^2-4c^3x+d^4# where #a,b,c,d# are positive real numbers. Prove that if #f# has four positive distinct roots, then #a > b > c > d#?

##### 2 Answers

#### Answer:

See below.

#### Explanation:

Supposing that

roots of

equating coefficients we have

We let to the reader as an exercise, to proof that if

As the number of positive roots is given as 4, the maximum number

of changes in signs of the coefficients has to be 4.

If either a or c or both are negative, the number of changes in sign

would become two or none So, the sufficient conditions for having

all four roots as positive is { a > 0 and c > 0 }.

This is more relaxed than the given conditions.