Step by step
The question
Question: write a polynomial with zeros −2 and 3i.
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 −2 gives x − (−2).
Minus a minus is plus: x + 2.
A complex zero always comes with its partner: the same number with the sign of the i part turned.
So −3i is a zero too.
Careful!
Don't forget the partner −3i.
Without it, the numbers in f(x) would not be real.
The pair 3i and −3i 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 3i squared.
(3i) squared is 9 times i squared: 9 times −1.
Minus a minus is plus: x² + 9.
No i is left!
2 Write f(x)
So f(x) = (x + 2)(x² + 9).
3 Multiply out
Each part of the first bracket times each part of the second.
(x + 2) times (x² + 9).
Each part times each part.
Like terms together: x³ + 2x² + 9x + 18.
So f(x) = x³ + 2x² + 9x + 18.
4 Check
Let's check: put each zero in.
First x = −2.
We get 0.
The pair: x² + 9 = 0 gives x squared is −9.
So x is plus or minus 3i.
It works!
Let's look back.
First: each zero gives a bracket.
Then: write f(x).
Then: multiply out.
Last: check.