Chapter 5, Problem 10CCSR

Solution

To Find : The number of real roots of the given cubic equation whose one root is given.

The number of real roots of the given cubic equation is $1$ .

Given information :

One root of a cubic equation is $2i$ .

Concept used :

Conjugate Root Theorem: If $P\left(x\right)$ is a polynomial with real coefficients, and $a+bi$ is one of its roots with $a\text{ and }b$ real numbers, then its complex conjugate $a-bi$ will also be its root.

Fundamental Theorem: If the degree of the polynomial is $n$ , which is more than or equal to one, then the polynomial will have exactly $n$ number of roots including real and complex roots.

Explanation :

Here, one of the roots of a cubic equation is $2i$ , so according to the Conjugate Root Theorem, $-2i$ will also be its root.

According to the Fundamental Theorem, the cubic equation will have total three roots.

So, there will be only one real root.