Polynomial with zeros 0, 3 and -2 Step by Step

Step by step

The question

Question: write a polynomial with zeros 0, 3 and −2.

zeros : 0, 3, −2

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

zero: an x where f(x) = 0

1 Each zero gives a bracket

If c is a zero, then x minus c is a factor.

Put c in, and that bracket is 0, so all of f(x) is 0.

zero c → bracket (x − c)

The zero 0 gives x − 0.

x = 0 → x − 0

Next, 3 gives x − 3.

x = 3 → x − 3

Next, −2 gives x − (−2).

Minus a minus is plus: x + 2.

x = −2 → x − (−2) = x + 2

Careful!

The sign flips.

The zero 3 gives x − 3, because that is 0 when x is 3.

Careful: not zero 3 → (x + 3) ✗

2 Write f(x)

So f(x) = (x)(x − 3)(x + 2).

f(x) = (x)(x − 3)(x + 2)

3 Multiply out

Each part of the first bracket times each part of the second.

each part × each part

(x) times (x − 3).

Each part times each part.

= x² − 3x(x + 2)

Like terms together: x² − 3x.

= (x² − 3x)(x + 2)

Now that bracket times the next one.

Each part times each part.

= x³ + 2x² − 3x² − 6x

Like terms together: x³ − x² − 6x.

= x³ − x² − 6x

So f(x) = x³ − x² − 6x.

f(x) = x³ − x² − 6x

4 Check

Let's check: put each zero in.

First x = 0.

We get 0.

f(0) = 0³ − 0² − 6(0) = 0 + 0 + 0 = 0

And x = 3.

We get 0.

f(3) = 3³ − 3² − 6(3) = 27 − 9 − 18 = 0

Let's look back.

First: each zero gives a bracket.

Then: write f(x).

Then: multiply out.

Last: check.

Answer