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

Step by step

The question

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

f(x) = (x − 3)(x + 1)²

1 One bracket is 0

Careful!

x + 1 is 0 when x is −1.

The sign changes.

Careful: not x + 1 = 0 → x = 1 ✗

x − 3 = 0 when x = 3.

x − 3 = 0 → x = 3

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 − 3) has power 1.

Odd: the graph crosses at 3.

x = 3 : power 1 → 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)² has power 2.

Even: the graph touches at −1.

x = −1 : power 2 → touches

So the zeros are 3, crossing and −1, touching.

3 See it on the graph

Draw the grid.

The y numbers go up in 2s.

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

At 3 it crosses the axis.

At −1 it touches the axis and turns back.

4 Check

Let's check: put 3 in.

One bracket is 0, so the answer is 0.

f(3) = (3 − 3)(3 + 1)² = 0

And −1 gives 0 too.

f(−1) = (−1 − 3)(−1 + 1)² = 0

Let's look back.

First: one bracket is 0.

Then: count the power.

Then: see it on the graph.

Last: check.

Answer