Step by step
The question
Question: write a polynomial with zeros 1, 2i and −2i.
A zero of f is an x that makes f(x) equal 0.
1 Each zero gives a bracket
If c is a zero, then x minus c is a factor.
Put c in, and that bracket is 0, so all of f(x) is 0.
The zero 1 gives x − 1.
Careful!
The sign flips.
The zero 1 gives x − 1, because that is 0 when x is 1.
The pair 2i and −2i gives two brackets.
A minus B, times A plus B, is A squared minus B squared.
And i squared is −1.
So we get x squared, minus 2i squared.
(2i) squared is 4 times i squared: 4 times −1.
Minus a minus is plus: x² + 4.
No i is left!
2 Write f(x)
So f(x) = (x − 1)(x² + 4).
3 Multiply out
Each part of the first bracket times each part of the second.
(x − 1) times (x² + 4).
Each part times each part.
Like terms together: x³ − x² + 4x − 4.
So f(x) = x³ − x² + 4x − 4.
4 Check
Let's check: put each zero in.
First x = 1.
We get 0.
The pair: x² + 4 = 0 gives x squared is −4.
So x is plus or minus 2i.
It works!
Let's look back.
First: each zero gives a bracket.
Then: write f(x).
Then: multiply out.
Last: check.