Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.

4, -8, and 2 + 3i

Answer 1

Option 3

`f(x) = x^4 - 35x^2+180x -416`

Explanation:

To answer this question we start by writing the polynomial product form of factors:

`(x-4)(x + 8)(x- (2 + 3i))(x- (2-3i))`

We multiply the first two factors:

`(x^2 + 8x -4x -32)(x- (2 + 3i))(x- (2-3i))\\\\(x^2 + 4x -32)(x- (2 + 3i))(x- (2-3i))`

Now we multiply the second two factors:

`(x^2+ 4x -32) (x^2 -2x+ 3ix -2x -3ix + 4 - 9i^2)`

We know that
`i = √((-1))`

So:

`i^2 = -1`

`(x^2 + 4x -32) (x^2 -4x + 4 + 9)\\\\(x^2 + 4x -32) (x^2 -4x + 13)`

Finally we multiply both terms and obtain the polynomial sought:

`(x^4 -4x^3 + 13x^2 + 4x^3 -16x^2 +52x-32x^2+128x -416)\\\\x^4 - 35x^2+180x -416`

Finally the correct option is the third.

`f(x) = x^4 - 35x^2+180x -416`