Question

Write a polynomial f(x) that satisfies the given conditions. Express the polynomial with the lowest possible leading positive

integer coefficient.
Polynomial of lowest degree with lowest possible integer coefficients, and with zeros 3-2i and 0 (multiplicity 4).

Answer 1

Explanation:

Let use the following rules to construct a polynomial,

Rule 1:

If r is a zero of some polynomial, f(x), then

`(x - r)`

is a factor of the polynomial

We first know that one of our zeroes are real, which is 0.

So since r=0

`(x - 0) = x`

is a factor

However, since the root ,0, has a multiplicty of 4, that basically means we have 0 as a root 4 times or basically we multiply the factor 4 times.

`x * x * x * x = x {}^(4)`

This leads to an another rule:

If a polynomial has a zero, r that has a multiplicty of k,

we represent the factor as

`(x - r) {}^(k)`

where k is all integers greater than 0.

So to reinforce myself, since 0 has a multiplicty of 4, we have

`(x - 0) {}^(4) = {x}^(4)`

So our first factor is

`{x}^(4)`

Part 2: Complex Zeroes.

When dealing with complex zeroes, we must note this rule.

if (a+bi) is a zero of some polynomial, p then its conjugate (a-bi) is a factor as well.

So if 3-2i is a factor, then

3+2i is a factor as well.

Using Rule 1, our factors now become

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

and

`(x - (3 - 2i))`

So our factors are

`{x}^(4) (x - (3 - 2i)(x - (3 + 2i))`

To simplify use FOIL Method,

`{x}^(4) ( {x}^(2) - x(3 + 2i) - x(3 - 2i) + 9 - 4 {i}^(2) )`

`{x}^(4) ( {x}^(2) - 6x + 13)`

`{x}^(6) - 6 {x}^(5) + 13 {x}^(4)`

So our polynomial is

`{x}^(6) - 6 {x}^(5) + 13 {x}^(4)`