Zeros of f(x) = (x + 2)^3(x - 1) Step by Step

Step by step

The question

Question: find the zeros of f(x) = (x + 2)³(x − 1).

f(x) = (x + 2)³(x − 1)

1 One bracket is 0

Careful!

x + 2 is 0 when x is −2.

The sign changes.

Careful: not x + 2 = 0 → x = 2 ✗

x + 2 = 0 when x = −2.

x + 2 = 0 → x = −2

f(x) is 0 when one of the brackets is 0.

Two numbers multiply to 0 only if one of them is 0.

A × B = 0 → A = 0 or B = 0

x − 1 = 0 when x = 1.

x − 1 = 0 → x = 1

2 Count the power

(x + 2)³ has power 3.

Odd: the graph crosses at −2.

x = −2 : power 3 → crosses

The power of each bracket is called the multiplicity of that zero.

Odd: the graph crosses the x-axis there.

Even: it only touches the axis and turns back.

multiplicity: the power of its bracket

odd: crosses the x-axis

even: touches it and turns back

(x − 1) has power 1.

Odd: the graph crosses at 1.

x = 1 : power 1 → crosses

So the zeros are −2, crossing and 1, crossing.

3 See it on the graph

Draw the grid.

Here is the graph of f(x) = (x + 2)³(x − 1).

At −2 it crosses the axis.

At 1 it crosses the axis.

4 Check

Let's check: put −2 in.

One bracket is 0, so the answer is 0.

f(−2) = (−2 + 2)³(−2 − 1) = 0

And 1 gives 0 too.

f(1) = (1 + 2)³(1 − 1) = 0

Let's look back.

First: one bracket is 0.

Then: count the power.

Then: see it on the graph.

Last: check.

Answer