Step by step
The question
Question: write a polynomial with zeros 4 and 1 + 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 4 gives x − 4.
A complex zero always comes with its partner: the same number with the sign of the i part turned.
So 1 − 2i is a zero too.
Careful!
Don't forget the partner 1 − 2i.
Without it, the numbers in f(x) would not be real.
The pair 1 + 2i and 1 − 2i gives two brackets.
Group x − 1 together in each bracket.
Then they look like A minus B, times A plus B.
A minus B, times A plus B, is A squared minus B squared.
And i squared is −1.
So we get (x − 1) squared, minus 2i squared.
(2i) squared is 4 times i squared: 4 times −1.
Minus a minus is plus: x² − 2x + 5.
No i is left!
2 Write f(x)
So f(x) = (x − 4)(x² − 2x + 5).
3 Multiply out
Each part of the first bracket times each part of the second.
(x − 4) times (x² − 2x + 5).
Each part times each part.
Like terms together: x³ − 6x² + 13x − 20.
So f(x) = x³ − 6x² + 13x − 20.
4 Check
Let's check: put each zero in.
First x = 4.
We get 0.
The pair: put x = 1 + 2i into x² − 2x + 5.
Remember i² = −1.
The i parts cancel, and the numbers make 0.
It works!
Let's look back.
First: each zero gives a bracket.
Then: write f(x).
Then: multiply out.
Last: check.