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

Step by step

The question

Question: find the zeros of f(x) = −2x²(x + 3).

f(x) = −2x²(x + 3)

1 One bracket is 0

Careful!

x + 3 is 0 when x is −3.

The sign changes.

Careful: not x + 3 = 0 → x = 3 ✗

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 + 3 = 0 when x = −3.

x + 3 = 0 → x = −3

2 Count the power

x² has power 2.

Even: the graph touches at 0.

x = 0 : power 2 → touches

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 + 3) has power 1.

Odd: the graph crosses at −3.

x = −3 : power 1 → crosses

So the zeros are 0, touching and −3, crossing.

3 See it on the graph

Draw the grid.

Here is the graph of f(x) = −2x²(x + 3).

At 0 it touches the axis and turns back.

At −3 it crosses the axis.

4 Check

Let's check: put 0 in.

One bracket is 0, so the answer is 0.

f(0) = −20²(0 + 3) = 0

And −3 gives 0 too.

f(−3) = −2(−3)²(−3 + 3) = 0

Let's look back.

First: one bracket is 0.

Then: count the power.

Then: see it on the graph.

Last: check.

Answer