Zeros of f(x) = x(x - 4)^2 Step by Step

Step by step

The question

Question: find the zeros of f(x) = x(x − 4)².

f(x) = x(x − 4)²

1 One bracket is 0

x = 0 is a zero too.

x = 0

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 − 4 = 0 when x = 4.

x − 4 = 0 → x = 4

2 Count the power

x has power 1.

Odd: the graph crosses at 0.

x = 0 : 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 − 4)² has power 2.

Even: the graph touches at 4.

x = 4 : power 2 → touches

So the zeros are 0, crossing and 4, 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(x − 4)².

At 0 it crosses the axis.

At 4 it touches the axis and turns back.

4 Check

Let's check: put 0 in.

One bracket is 0, so the answer is 0.

f(0) = 0(0 − 4)² = 0

And 4 gives 0 too.

f(4) = 4(4 − 4)² = 0

Let's look back.

First: one bracket is 0.

Then: count the power.

Then: see it on the graph.

Last: check.

Answer