Step by step
The question
Question: find all the zeros of f(x) = x³ − x² − 10x − 8.
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 −8.
Its factors are 1, 2, 4, 8, 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 −18, not 0.
So 1 is not a zero.
Next −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, −1, −10, −8.
The first number comes straight down: 1.
1 times −1 is −1.
Write it under the next number.
Add: −1 + (−1) is −2.
−2 times −1 is 2.
Write it under the next number.
Add: −10 + 2 is −8.
−8 times −1 is 8.
Write it under the next number.
Add: −8 + 8 is 0.
The last number is the remainder: 0.
The other numbers are the answer's numbers: 1, −2, −8.
The answer starts one power lower than x³.
So f(x) = (x + 1)(x² − 2x − 8).
The other zeros come from x² − 2x − 8 = 0.
4 Solve the quadratic
Now x² − 2x − 8 = 0, with the quadratic lesson.
5 Factor
We need two numbers.
They multiply to −8, and they add to −2.
They multiply to a minus number, so one number is plus and one is minus.
They add to a minus number, so the bigger one is minus.
We list the pairs that make 8, smallest first.
We take the smaller from the bigger.
We need 2.
2 and 4 give 2.
Yes!
With the signs: 2 and −4.
So it is (x + 2)(x − 4) = 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 take away 2 on both sides.
In x − 4 = 0, we add 4 on both sides.
So x = −2 or x = 4.
7 Write all the zeros
So the zeros are −1, −2 and 4.
−1 from the test, the others from the quadratic.
8 Check
Let's check: put −2 into f.
We get 0.
And put 4 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.