Zeros, roots and factors | Year 12

01 / LEARN

Understand the idea.

Three descriptions of the same event

A zero of $P$ is a number $a$ satisfying $P (a) = 0$ . The same number is a root of the equation $P (x) = 0$ . By the factor theorem, the corresponding linear factor is $x - a$ .

The sign in a factor is opposite to its root: $x + 2 = x - (- 2)$ vanishes at $- 2$ . A root is a number and a factor is a polynomial expression; keep these two types of answer distinct.

Repeated roots and the graph

If ${(x - a)}^{m}$ is the highest power of that factor dividing $P$ , the root has multiplicity $m$ . A repeated root is still one distinct input, although it contributes more than one to the polynomial degree.

Near an isolated real root, odd multiplicity changes the sign and crosses the axis; even multiplicity keeps the sign and touches the axis. Other nonzero factors keep their sign nearby. Use this local fact alongside intercepts and end behaviour.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Reading zeros from factors

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

Worked solution

1
Find the zero of each linear factor.
$\begin{aligned} x + 2 = 0 & \Rightarrow x = - 2, \\ x - 1 = 0 & \Rightarrow x = 1. \end{aligned}$
A product is zero when at least one of its factors is zero.
2
Read the multiplicities.
$P (x) = {(x + 2)}^{3} {(x - 1)}^{2} .$
The factor powers give multiplicity three at $- 2$ and multiplicity two at $1$ .
3
Interpret the local graph behaviour.
$\begin{aligned} x = - 2 & : crosses the axis, \\ x = 1 & : touches the axis . \end{aligned}$
Odd multiplicity changes sign; even multiplicity does not. The multiplicities total the degree five.

Answer

$- 2$ (multiplicity $3$ ), $1$ (multiplicity $2$ )

EXAMPLE 02

Checking a proposed root

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

Worked solution

1
Evaluate the proposed root.
$P (2) = 2^{3} - 3 (2^{2}) + 4 = 8 - 12 + 4 = 0.$
A root must make the original polynomial equal to zero.
2
Apply the factor theorem.
$P (2) = 0 \Rightarrow x - 2 is a factor of P (x) .$
The factor associated with root $a$ is $x - a$ .
3
Confirm with a factorisation.
$\begin{aligned} P (x) & = (x - 2) (x^{2} - x - 2) \\ = {(x - 2)}^{2} (x + 1) . \end{aligned}$
Expansion reproduces the original cubic and shows that this root is actually repeated.

Answer

Yes; $x - 2$ is a factor.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

One, $x = 4$ .

The equation ${(x - 4)}^{2} = 0$ forces $x - 4 = 0$ , so its only distinct real zero is $4$ . The exponent two gives that one zero multiplicity two.

Keep practising

6 worksheet questions · 12 marks

Open worksheet 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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Curriculum reference & lesson scope

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.

Official curriculum source ↗

In this lesson

Understand the idea.

Three descriptions of the same event

Repeated roots and the graph

See the method in action.

Reading zeros from factors

Find the zero of each linear factor.

Read the multiplicities.

Interpret the local graph behaviour.

Checking a proposed root

Evaluate the proposed root.

Apply the factor theorem.

Confirm with a factorisation.

Try it yourself.

Check your understanding

Keep practising