Step by step
The question
Question: find all the zeros of f(x) = x³ − 6x² + 11x − 6.
A zero of f is an x that makes f(x) equal 0.
A power 3 has up to three zeros.
1 List the candidates
The rational root theorem: a whole or fraction zero is a factor of the last number, over a factor of the first number.
The last number is −6.
Its factors are 1, 2, 3, 6, each with plus and minus.
Careful!
Don't forget the negative ones.
−1, −2 and the others can be zeros too.
The first number is 1.
So the candidates are just these.
2 Test the candidates
We put each candidate into f.
When we get 0, we found a zero.
Then x minus that number is a factor, like in the factor theorem lesson.
Start with 1: put it in place of every x.
We get 0.
So x = 1 is a zero!
3 Divide by x − 1
x = 1 is a zero, so (x − 1) is a factor.
We divide it out, with synthetic division.
x − 1 is 0 when x is 1.
So we use 1.
Then we write just the numbers in front: 1, −6, 11, −6.
The first number comes straight down: 1.
1 times 1 is 1.
Write it under the next number.
Add: −6 + 1 is −5.
−5 times 1 is −5.
Write it under the next number.
Add: 11 + (−5) is 6.
6 times 1 is 6.
Write it under the next number.
Add: −6 + 6 is 0.
The last number is the remainder: 0.
The other numbers are the answer's numbers: 1, −5, 6.
The answer starts one power lower than x³.
So f(x) = (x − 1)(x² − 5x + 6).
The other zeros come from x² − 5x + 6 = 0.
4 Solve the quadratic
Now x² − 5x + 6 = 0, with the quadratic lesson.
5 Factor
We need two numbers.
They multiply to 6, and they add to −5.
They multiply to a plus number, so the two numbers have the same sign.
They add to a minus number, so both are minus.
We list the pairs that make 6, smallest first.
We add each pair.
We need 5.
2 and 3 give 5.
Yes!
With the signs: −2 and −3.
So it is (x − 2)(x − 3) = 0.
6 One bracket is 0
Two numbers multiply to 0.
That only happens if one of them is 0.
So one bracket must be 0.
We make each bracket 0.
Two small equations.
In x − 2 = 0, we add 2 on both sides.
In x − 3 = 0, we add 3 on both sides.
So x = 2 or x = 3.
7 Write all the zeros
So the zeros are 1, 2 and 3.
1 from the test, the others from the quadratic.
8 Check
Let's check: put 2 into f.
We get 0.
And put 3 into f.
We get 0 too.
Let's look back.
First: list the candidates.
Then: test the candidates.
Then: divide by x − 1.
Then: solve the quadratic.
Then: factor.
Then: one bracket is 0.
Then: write all the zeros.
Last: check.