Question

Write a polynomial function of least degree with integral coefficients that have the given zeros.

Answer 1

Using the Factor Theorem, it is found that the polynomial function is given by:

• $f(x) = a(x^4 - 3x^3 - x^2 - 27x - 90)$, in which a is the leading coefficient.

Factor Theorem:

• The Factor Theorem states that a polynomial function with roots $x_1, x_2, \codts, x_n$ is given by:

$f(x) = a(x - x_1)(x - x_2) \cdots (x - x_n)$

• In which a is the leading coefficient.

In this problem, we have that there is a:

• Zero at x = -2, hence $x_1 = -2$.

• Zero at x = 5, hence $x_2 = 5$.

• Zero at x = -3i, hence there also has to be a zero at it's conjugate x = 3i, hence $x_3 = -3i, x_4 = 3i$.

Then, the function is:

$f(x) = a(x + 2)(x - 5)(x + 3i)(x - 3i)$

$f(x) = a(x^2 - 3x - 10)(x^2 + 9)$

$f(x) = a(x^4 - 3x^3 - x^2 - 27x - 90)$

To learn more about the Factor Theorem, you can take a look at brainly.com/question/24380382