Answer:
The required polynomial is [tex] x^3-2x^2+x-2[/tex]
Step-by-step explanation:
Given : A polynomial equation of degree 3 such that two of its roots are 2 and an imaginary number.
To find : The equation of polynomial with degree 3.
Solution :
It is given that the equation has 3 roots one is 2 and othe is imaginary.
So, one root 2 = (x-2)
Let the other two roots are imaginary i, -i
⇒ (x-i),(x+i)
Therefore, the roots of the polynomial of degree 3
[tex](x-2)(x-i)(x+i)[/tex]
Now, we solve the roots to find the equation,
[tex]\Rightarrow(x-2)(x^2+xi-xi-i^2)[/tex]
[tex]\Rightarrow(x-2)(x^2+i^2)[/tex] [tex][i^2=-1][/tex]
[tex]\Rightarrow(x^3+x-2x^2-2)[/tex]
[tex]\Rightarrow x^3-2x^2+x-2[/tex]
Therefore, the required polynomial is [tex] x^3-2x^2+x-2[/tex]
