Question

The number of real zeroes, a cubic polynomial cannot have

• A. 3
• B. 2
• C. 1
• D. 0

Answer 1

The correct option is D 0

Consider a cubic polynomial with all its coefficients as real numbers ${\mathrm{ax}}^3+{\mathrm{bx}}^2+\mathrm{cx}+d,a\ne 0.$

Let $\alpha ,\beta ,\gamma$ be its zeroes. Then,

$\mathrm{\alpha \beta \gamma }=\frac{-d}{a}\in R$

Case I: If all zeroes are non-real, then product of zeroes is also non-real, which is not possible as

$\frac{-d}{a}\in R$

Case II: If one zero is real and others are non-real, then the product of zeroes is real as the product of 2 non-real numbers is real.

Therefore, a cubic polynomial can have atleast one real zero.

Hence, the correct answer is option (d).