ISI2014-DCG-30

Question

Consider the equation $P(x) =x^3+px^2+qx+r=0$ where $p,q$ and $r$ are all real and positive. State which of the following statements is always correct.

• A. All roots of $P(x) = 0$ are real
• B. The equation $P(x)=0$ has at least one real root
• C. The equation $P(x)=0$ has no negative real root
• D. The equation $P(x)=0$ must have one positive and one negative real root

Answer 1

Every cubic equation must have atleast one real root, as because only two cases are possible for the roots of a cubic equation

1. All 3 roots are real
2. One real root and two non-real complex conjugate roots

Thus Option(B) is always correct.

Now,

For Option(A) -> Complex roots are also possible hence false

For Option(C) -> Since the coefficients $p,q$ and $r$ are all positive, $\alpha \beta \gamma =-r$ is negative (where $\alpha , \beta$ and $\gamma$ are the three roots of the cubic equation) , thus atleast one root has to be negative to make the product of the three roots negative.Hence false.

For Option (D) As stated above if all three roots are negative then also the product of three roots will be negative.Hence false. 

For derving the formulae $\alpha \beta \gamma =-r$ , check Vieta's formulas.