Step by step
The question
Question: find all the zeros of f(x) = x³ − 3x² − 10x + 24.
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 24.
Its factors are 1, 2, 3, 4, 6, 8, 12, 24, 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 12, not 0.
So 1 is not a zero.
Next −1: put it in place of every x.
We get 30, not 0.
So −1 is not a zero.
Next 2: put it in place of every x.
We get 0.
So x = 2 is a zero!
3 Divide by x − 2
x = 2 is a zero, so (x − 2) is a factor.
We divide it out, with synthetic division.
x − 2 is 0 when x is 2.
So we use 2.
Then we write just the numbers in front: 1, −3, −10, 24.
The first number comes straight down: 1.
1 times 2 is 2.
Write it under the next number.
Add: −3 + 2 is −1.
−1 times 2 is −2.
Write it under the next number.
Add: −10 + (−2) is −12.
−12 times 2 is −24.
Write it under the next number.
Add: 24 + (−24) 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 − 2)(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 2, −3 and 4.
2 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 − 2.
Then: solve the quadratic.
Then: factor.
Then: one bracket is 0.
Then: write all the zeros.
Last: check.