Write a polynomial function of least degree that has the following zeros. 4 and I

Answer 1

Hi Kelvin,

The function must have at least degree three, one for each of the zeros. Even though only two zeros were given, the imaginary ones (like i in this problem) always come in pairs so -i will also be a zero.

The way we build a function is by turning each zero into a factor. For example, 4 becomes (x-4).

That means we have f(x) = (x-i)(x-(-i))(x-4) as a function.

Note that x-(-i) is x+i so we have

f(x) = (x - i)(x + i))(x - 4)

If we expand it by distributing the parentheses:

f(x) = (x² + ix - ix - i²)(x - 4) Note that the ix terms cancel and i² = -1 by definition.

f(x) = (x² - (-1))(x - 4)

f(x) = (x² + 1)(x - 4)

f(x) = x³ - 4x² + x - 4

Please let me know if I can be more helpful, thanks!

Joe

Answer 2

If p(x) is a polynomial with x = i as a zero, then x = -i is also a zero. And we are also given x = 4 as a zero. Therefore the smallest possible degree for p(x) is 3 and one such polynomial is

p(x) = (x - i)(x +i)(x -4)

= (x² + 1)(x - 4)

= x³ - 4x² + x - 4