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