Step by step
The question
Question: write a polynomial with zeros −1, 2 and 4.
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).
Minus a minus is plus: x + 1.
Next, 2 gives x − 2.
Next, 4 gives x − 4.
Careful!
The sign flips.
The zero −1 gives x + 1, because that is 0 when x is −1.
2 Write f(x)
So f(x) = (x + 1)(x − 2)(x − 4).
3 Multiply out
Each part of the first bracket times each part of the second.
(x + 1) times (x − 2).
Each part times each part.
Like terms together: x² − x − 2.
Now that bracket times the next one.
Each part times each part.
Like terms together: x³ − 5x² + 2x + 8.
So f(x) = x³ − 5x² + 2x + 8.
4 Check
Let's check: put each zero in.
First x = −1.
We get 0.
And x = 2.
We get 0.
Let's look back.
First: each zero gives a bracket.
Then: write f(x).
Then: multiply out.
Last: check.