Class 10 Math All topics Polynomials

If all three zeroes of a cubic polynomial $x_{3}+ax_{2}+bx+c$ are positive, then which of the following is correct about $a,b$ and $c$?

(a)

$b$ must be positive

(b)

$a$ must be positive

(c)

$c$ must be positive

(d)

$b$ must be negative

Correct answer: (a)

Solution: Let the roots of the equation $x_{3}+ax_{2}+bx+c=0$ be $α,β,γ$

Then, sum of roots, $α+β+γ=−a$. Thus, $a$ is negative.

Product of roots, taken two at a time, $α.β+α.γ+γ.β=b$. Thus $b$ is positive

Product of roots, $α.β.γ=−c$. Thus $c$ is negative.

Hence, $a$ and $c$ have -ve sign, but $b$ has +ve sign.

Then, sum of roots, $α+β+γ=−a$. Thus, $a$ is negative.

Product of roots, taken two at a time, $α.β+α.γ+γ.β=b$. Thus $b$ is positive

Product of roots, $α.β.γ=−c$. Thus $c$ is negative.

Hence, $a$ and $c$ have -ve sign, but $b$ has +ve sign.

Similar topics

introduction to trigonometry

functions

some applications of trigonometry

quadratic equations

surface areas and volumes

introduction to trigonometry

functions

some applications of trigonometry

quadratic equations

surface areas and volumes