Step by step
The question
Question: find all the zeros of f(x) = x³ − 13x − 12.
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 −12.
Its factors are 1, 2, 3, 4, 6, 12, 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 −24, 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, 0, −13, −12.
There is no x² part.
We write 0x², so the columns line up.
The first number comes straight down: 1.
1 times −1 is −1.
Write it under the next number.
Add: 0 + (−1) is −1.
−1 times −1 is 1.
Write it under the next number.
Add: −13 + 1 is −12.
−12 times −1 is 12.
Write it under the next number.
Add: −12 + 12 is 0.
The last number is the remainder: 0.
The other numbers are the answer's numbers: 1, −1, −12.
The answer starts one power lower than x³.
So f(x) = (x + 1)(x² − x − 12).
The other zeros come from x² − x − 12 = 0.
4 Solve the quadratic
Now x² − x − 12 = 0, with the quadratic lesson.
5 Factor
We need two numbers.
They multiply to −12, and they add to −1.
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 12, smallest first.
We take the smaller from the bigger.
We need 1.
3 and 4 give 1.
Yes!
With the signs: 3 and −4.
So it is (x + 3)(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 + 3 = 0, we take away 3 on both sides.
In x − 4 = 0, we add 4 on both sides.
So x = −3 or x = 4.
7 Write all the zeros
So the zeros are −1, −3 and 4.
−1 from the test, the others from the quadratic.
8 Check
Let's check: put −3 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.