Zeros, roots and factors | Year 12

01 / LEARN

The idea, step by step

Three descriptions of the same event

A zero of $P$ is an input $a$ for which $P (a) = 0$ . It is a root of the equation $P (x) = 0$ , and $x - a$ is a factor of $P (x)$ . The sign matters: a root of $- 3$ corresponds to the factor $x + 3$ . A root is a number; a factor is an expression.

A factored form gives zeros efficiently. For $P (x) = (x - 2) {(x + 1)}^{2}$ , the distinct real zeros are $2$ and $- 1$ . Counting multiplicity, the zero $- 1$ appears twice and $2$ once. Distinct zeros and degree therefore need not be equal.

Repeated roots and the graph

Near a root, the factors that are non-zero keep a fixed sign. A factor ${(x - a)}^{k}$ changes sign across $a$ when $k$ is odd, but not when $k$ is even. An odd-multiplicity real root crosses the axis; an even-multiplicity root touches and turns back.

This crossing rule does not locate every turning point and does not replace a full sketch. Use it together with degree, leading coefficient and intercepts. A real polynomial of degree $n$ has at most $n$ distinct real zeros; it may have fewer, including none when its degree is even.

A common trap

A zero of $- 2$ means the factor is $x - 2$ .

Instead: The factor is $x - (- 2) = x + 2$ . Substitute the root to check that the factor is zero.

02 / WORKED EXAMPLES

See the thinking

Example 1

Reading zeros from factors

Find the real zeros and their multiplicities for $P (x) = {(x + 2)}^{3} {(x - 1)}^{2}$ .

$x + 2 = 0$ gives $x = - 2$ with multiplicity $3$ .
$x - 1 = 0$ gives $x = 1$ with multiplicity $2$ . The graph crosses at $- 2$ and touches at $1$ .

- 2

(multiplicity

3

1

(multiplicity

2

)

Example 2

Checking a proposed root

Is $2$ a root of $P (x) = x^{3} - 3 x^{2} + 4$ ? What factor follows?

$P (2) = 8 - 12 + 4 = 0$ .
Therefore $2$ is a root and $x - 2$ is a factor. Indeed $P (x) = {(x - 2)}^{2} (x + 1)$ .

Yes;

x - 2

is a factor.

Check your understanding

How many distinct real zeros does ${(x - 4)}^{2}$ have?

Show answer

One, $x = 4$ .

Multiplicity two does not create a second distinct number.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

List the zeros of $(x - 5) (x + 2) (2 x - 3)$ .
2 marks
State multiplicities and crossing behaviour for $x^{2} {(x + 1)}^{3}$ .
2 marks
Check whether $- 1$ is a zero of $x^{3} + 2 x^{2} - x - 2$ .
2 marks
Write a monic cubic with zeros $- 2, 1, 4$ .
2 marks
Give a quadratic with no real zeros and explain how you know.
2 marks
A quartic has three distinct zeros, one of them repeated twice and the others simple. How many roots are counted with multiplicity?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

List the zeros of $(x - 5) (x + 2) (2 x - 3)$ .
2 marks
Set each factor equal to zero.
$5, - 2, \frac{3}{2}$
State multiplicities and crossing behaviour for $x^{2} {(x + 1)}^{3}$ .
2 marks
Even powers preserve sign; odd powers change sign.
$0$ has multiplicity $2$ and touches; $- 1$ has multiplicity $3$ and crosses.
Check whether $- 1$ is a zero of $x^{3} + 2 x^{2} - x - 2$ .
2 marks
$P (- 1) = - 1 + 2 + 1 - 2 = 0$ .
Yes.
Write a monic cubic with zeros $- 2, 1, 4$ .
2 marks
The linear factors give each root; monic means leading coefficient $1$ .
$(x + 2) (x - 1) (x - 4) = x^{3} - 3 x^{2} - 6 x + 8$
Give a quadratic with no real zeros and explain how you know.
2 marks
For real $x$ , $x^{2} \geq 0$ , so $x^{2} + 1 > 0$ .
For example $x^{2} + 1$ .
A quartic has three distinct zeros, one of them repeated twice and the others simple. How many roots are counted with multiplicity?
2 marks
$2 + 1 + 1 = 4$ . The repeated zero is still one distinct input.
Four.

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SACE Board of South Australia

A recognisable real-polynomial chapter supporting Stage 2 Specialist polynomial factorisation and graph analysis; original lessons use current SACE scope and the school chapter structure requested by the user.

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