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

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

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`