Question

I think the answer is A. Can you check if I am correct in my thinking? Thanks!

Answer 1

Step 1:

In this question, we are given the following:

Step 2:

The details of the solution are as follows:

`\begin{gathered} Given\text{ the function,} \\ f(x)\text{ = x}^3-20x^2\text{ + 123 x - 216} \\ Using\text{ Rational Root theorem, we have that:} \\ f(x)\text{ =}\frac{x^3-20\text{ x}^2+\text{ 123x - 216}}{x-\text{ 3}}=\text{ x}^2-\text{ 17 x + 72} \\ Now,\text{ factorizing x}^2-17x\text{ + 72 , we have that:} \\ \end{gathered}`
`\begin{gathered} x^2\text{ -17 x+ 72 = \lparen x- 8 \rparen \lparen x - 9 \rparen} \\ Hence,\text{ we can see that:} \\ x^3-20x^2+\text{ 123 x - 216 = \lparen x- 3\rparen \lparen x - 8 \rparen \lparen x - 9 \rparen} \\ Hence,\text{ we have zero imaginary roots and 3 real roots} \end{gathered}`

CONCLUSION:

The final answer is:

`zero\text{ imaginary , 3 real roots \lparen OPTION A \rparen}`