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

Step by step

The question

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

1 One bracket is 0

Careful!

x + 1 is 0 when x is −1.

The sign changes.

x − 2 = 0 when 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.

x + 1 = 0 when x = −1.

2 Count the power

(x − 2)² has power 2.

Even: the graph touches at 2.

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.

(x + 1)³ has power 3.

Odd: the graph crosses at −1.

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

3 See it on the graph

Draw the grid.

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

At 2 it touches the axis and turns back.

At −1 it crosses the axis.

4 Check

Let's check: put 2 in.

One bracket is 0, so the answer is 0.

And −1 gives 0 too.

Let's look back.

First: one bracket is 0.

Then: count the power.

Then: see it on the graph.

Last: check.