Question

Which of the following is a polynomial with roots 2, 3i, and −3i? f(x) = x3 − 2x2 + 6x − 9 f(x) = x3 − 6x2 + 9x − 18 f(x) = x3 − 6x2 + 18x − 2 f(x) = x3 − 2x2 + 9x − 18

Answer 1

f(x) = x3 − 2x2 + 9x − 18

Answer 2

Answer:

$f(x)=x^3-2 x^2+9 x-18$

Step-by-step explanation:

The roots of the polynomial are $2,3i,-3i$.

This implies that $x-2,x-3i,x+3i$ are factors of the given polynomial.

The polynomial will have equation;

$f(x)=(x-2)(x-3i)(x+3i)$

We expand using difference of two squares on the complex conjugates to get;

$f(x)=(x-2)(x^2-(3i)^2)$

$\Rightarrow f(x)=(x-2)(x^2-(-3)^2(i)^2)$.

$\Rightarrow f(x)=(x-2)(x^2-9(i)^2)$.

Recall that;

$\boxed{i^2=-1}$

$\Rightarrow f(x)=(x-2)(x^2+9)$.

Expand using the distributive property to get;

$\Rightarrow f(x)=x^3+9x-2x^2-18$.

We rewrite in standard form to obtain;

$f(x)=x^3-2x^2+9 x-18$