Step by step
The question
Question: find all the zeros of f(x) = x³ + x² − 4x − 4.
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 −4.
Its factors are 1, 2, 4, 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 −6, 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, −4, −4.
The first number comes straight down: 1.
1 times −1 is −1.
Write it under the next number.
Add: 1 + (−1) is 0.
0 times −1 is 0.
Write it under the next number.
Add: −4 + 0 is −4.
−4 times −1 is 4.
Write it under the next number.
Add: −4 + 4 is 0.
The last number is the remainder: 0.
The other numbers are the answer's numbers: 1, 0, −4.
The answer starts one power lower than x³.
So f(x) = (x + 1)(x² − 4).
The other zeros come from x² − 4 = 0.
4 Solve the quadratic
Now x² − 4 = 0, with the quadratic lesson.
5 Get x² alone
There is no x term, so we get x² by itself.
To remove the − 4, we add 4 on both sides.
0 + 4 = 4.
Now we have x² = 4.
6 Square root of both sides
x squared is 4.
So x is the square root of 4, or minus it.
Minus times minus is plus, so the negative one works too.
2 times 2 is 4.
So x = 2 or x = −2.
7 Write all the zeros
So the zeros are −1, 2 and −2.
−1 from the test, the others from the quadratic.
8 Check
Let's check: put 2 into f.
We get 0.
And put −2 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: get x² alone.
Then: square root of both sides.
Then: write all the zeros.
Last: check.