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Find all zeros of f(x) = x^3 - 2x^2 - 9x + 18

Step by step, with the reason for every step · Finding all zeros · Algebra 2

Step by step

The question

Question: find all the zeros of f(x) = x³ − 2x² − 9x + 18.

f(x) = x3 − 2x2 − 9x + 18

A zero of f is an x that makes f(x) equal 0.

A power 3 has up to three zeros.

a zero: an x where f(x) = 0

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.

whole or fraction zero = factor of the last numberfactor of the first number

The last number is 18.

Its factors are 1, 2, 3, 6, 9, 18, each with plus and minus.

factors of 18 : ±1, ±2, ±3, ±6, ±9, ±18

Careful!

Don't forget the negative ones.

−1, −2 and the others can be zeros too.

Careful: every candidate has + and −

The first number is 1.

So the candidates are just these.

candidates : ±1, ±2, ±3, ±6, ±9, ±18

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.

f(c) = 0 → (x − c) is a factor

Start with 1: put it in place of every x.

f(1) = 13 − 2(1)2 − 9(1) + 18

We get 8, not 0.

So 1 is not a zero.

= 1 − 2 − 9 + 18 = 8 ✗

Next −1: put it in place of every x.

f(−1) = (−1)3 − 2(−1)2 − 9(−1) + 18

We get 24, not 0.

So −1 is not a zero.

= −1 − 2 + 9 + 18 = 24 ✗

Next 2: put it in place of every x.

f(2) = 23 − 2(2)2 − 9(2) + 18

We get 0.

So x = 2 is a zero!

= 8 − 8 − 18 + 18 = 0 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, −2, −9, 18.

The first number comes straight down: 1.

1 times 2 is 2.

Write it under the next number.

Add: −2 + 2 is 0.

0 times 2 is 0.

Write it under the next number.

Add: −9 + 0 is −9.

−9 times 2 is −18.

Write it under the next number.

Add: 18 + (−18) is 0.

The last number is the remainder: 0.

The other numbers are the answer's numbers: 1, 0, −9.

The answer starts one power lower than x³.

So f(x) = (x − 2)(x² − 9).

The other zeros come from x² − 9 = 0.

f(x) = (x − 2)(x² − 9)

4 Solve the quadratic

Now x² − 9 = 0, with the quadratic lesson.

x2 − 9 = 0

5 Get x² alone

There is no x term, so we get x² by itself.

To remove the − 9, we add 9 on both sides.

x2 − 9 + 9 = 0 + 9

0 + 9 = 9.

Now we have x² = 9.

x2 = 9

6 Square root of both sides

x squared is 9.

So x is the square root of 9, or minus it.

Minus times minus is plus, so the negative one works too.

x² = k → x = √k or x = −√k
3 × 3 = 9, (−3) × (−3) = 9

3 times 3 is 9.

So x = 3 or x = −3.

x = 3 or x = −3

7 Write all the zeros

So the zeros are 2, 3 and −3.

2 from the test, the others from the quadratic.

zeros : x = 2 or x = 3 or x = −3

8 Check

Let's check: put 3 into f.

We get 0.

x = 3: f = 33 − 2(3)2 − 9(3) + 18
= 27 − 18 − 27 + 18 = 0

And put −3 into f.

We get 0 too.

x = −3: f = (−3)3 − 2(−3)2 − 9(−3) + 18
= −27 − 18 + 27 + 18 = 0

Let's look back.

First: list the candidates.

Then: test the candidates.

Then: divide by x − 2.

Then: solve the quadratic.

Then: get x² alone.

Then: square root of both sides.

Then: write all the zeros.

Last: check.

Answer
zeros : x = 2 or x = 3 or x = −3

Your turn

Try this one yourself, then check:

find all zeros of f(x) = x3 − x2 − 9x + 9
Show the answer
x = 1 or x = 3 or x = −3

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