Year 12 Maths Extension 1 (2027) Further applications of calculus

Multiplicity of zeroes via calculus (P′ multiplicity, behaviour at roots)

Q: What is the multiplicity of a root?

It is how many times the factor x minus alpha appears in P of x.

Q: How does calculus find a multiple root?

A root of multiplicity greater than one satisfies both P of alpha equals zero and P prime of alpha equals zero, so solve them together.

Q: What does the graph look like at a double root?

The curve is tangent to the x-axis; it touches but does not cross.

Q: Why is alpha a root of P prime?

By the product rule, P prime of x has a factor of x minus alpha to the m minus one, so alpha is a zero of P prime of multiplicity m minus one.

20 practice questions 2 video lessons Theory + worked examples

Explore the multiplicity of zeroes using calculus in NSW Year 12 Mathematics Extension 1. A repeated root of a polynomial shows up in its derivative, revealing how the curve meets the x-axis.

You will learn to use the product rule to link a root's multiplicity in a polynomial to its multiplicity in the derivative, determine the multiplicity of a zero, and sketch how the curve behaves at each root — connecting algebra and calculus in Extension 1.

Practice 20 questions

Practice questions

Every question with a fully worked solution.

Start practising

Watch 2 video(s)

HSC 4U Maths: Polynomials - Roots of Multiplicity Example 1 Watch
How to determine the multiplicity of a zero or root of a polynomial by differentiation Watch

Theory

The multiplicity of a zero counts how often $(x - α)$ divides $P (x)$ . This NSW Year 12 Mathematics Extension 1 topic (NESA outcome ME1-12-05) links it to $P^{'} (x)$ — a multiple root satisfies $P (α) = 0$ and $P^{'} (α) = 0$ — and to whether the graph crosses, touches or inflects.

The multiplicity of a zero $α$ of a polynomial $P (x)$ is how many times the factor $(x - α)$ appears. Calculus links it to $P^{'} (x)$ and to the shape of the graph at $α$ .

If $α$ is a zero of multiplicity $m > 1$ , then $α$ is also a zero of $P^{'} (x)$ , of multiplicity $m - 1$ . So a multiple root satisfies both $P (α) = 0$ and $P^{'} (α) = 0$ .

The multiplicity fixes the shape at the intercept: $m = 1$ the curve crosses; $m > 1$ even it is tangent (touches, does not cross); $m \geq 3$ odd it has a horizontal inflection (crosses, flattening).

NESA link. Part of the Year 12 Further applications of calculus focus area, outcome ME1-12-05 ("applies calculus to solve problems involving polynomials, further rates of change, areas and volumes and differential equations") with MAO-WM-01. A syllabus example: if $x = 2$ is a double root of $P (x) = x^{3} + a x^{2} + b x - 4$ , find $a$ and $b$ .

Cross

(m = 1)

, touch

(m = 2)

, horizontal inflection

(m = 3)

y = (x - 1)^{2} (x + 2)

: touches at

x = 1

(even), crosses at

x = - 2

P (x) = (x - α)^{m} Q (x), Q (α) \neq 0 \Rightarrow P^{'} (x) = (x - α)^{m - 1} [m Q (x) + (x - α) Q^{'} (x)] .

if alpha has multiplicity m in P, it has multiplicity m-1 in P'

A multiple root therefore satisfies $P (α) = 0$ and $P^{'} (α) = 0$ together. Behaviour at the intercept:

Multiplicity $m$	Behaviour at $α$
$m = 1$	crosses the axis (no flattening)
$m > 1$ even	tangent — touches, does not cross
$m \geq 3$ odd	horizontal inflection — crosses, flattening

How to find or use a multiple root

Differentiate. Solve $P^{'} (x) = 0$ to find candidate multiple roots.
Test in $P$ . A candidate $α$ is a multiple root only if $P (α) = 0$ as well.
Factor. Write $P (x) = (x - α)^{m} Q (x)$ to read off the remaining roots.
Describe the graph using the multiplicity: cross, touch, or horizontal inflection.

Example 1 — Read the behaviour

Describe the behaviour of

y = (x - 1)^{2} (x + 3)^{3}

at each

x

-intercept.

Solution

At $x = 1$ (multiplicity $2$ , even): the curve is tangent to the axis and does not cross.

At $x = - 3$ (multiplicity $3$ , odd): the curve has a horizontal inflection and crosses.

x=1 touches (m=2 even); x=-3 horizontal inflection (m=3 odd)

Example 2 — Find the double root

Show

P (x) = x^{3} - 12 x + 16

has a multiple zero and find it.

Solution

Solve $P^{'} (x) = 0$ .

$P^{'} (x) = 3 x^{2} - 12$	$=$	$0 \Rightarrow x = \pm 2$
$P (2) = 8 - 24 + 16$	$=$	$0$

P(x) = (x-2)^2 (x+4); double root at x=2

So $x = 2$ is a double root and $P (x) = (x - 2)^{2} (x + 4)$ .

Example 3 — A parameter

P (x) = x^{3} - 3 x^{2} - 9 x + k

has a double root. Find the possible

k

Solution

$P^{'} (x) = 3 (x - 3) (x + 1) = 0$ , so $x = 3$ or $x = - 1$ .

$P (3) = 0$	$\Rightarrow$	$k = 27$
$P (- 1) = 0$	$\Rightarrow$	$k = - 5$

k = 27 (roots 3,3,-3) or k = -5 (roots -1,-1,5)

$k = 27$ (roots $3, 3, - 3$ ) or $k = - 5$ (roots $- 1, - 1, 5$ ).

Example 4 — The proof

P (x) = (x - α)^{m} Q (x)

with

m > 1

Q (α) \neq 0

, show

α

is a zero of

P^{'}

of multiplicity

m - 1

Solution

By the product rule:

P^{'} (x)

=

(x - α)^{m - 1} [m Q (x) + (x - α) Q^{'} (x)]

P'(x) = (x-alpha)^(m-1)[mQ(x)+(x-alpha)Q'(x)]

At $x = α$ the bracket is $m Q (α) \neq 0$ , so $α$ has multiplicity exactly $m - 1$ in $P^{'}$ .

Common pitfalls

Checking only $P^{'}$ . A multiple root needs both

P (α) = 0

and

P^{'} (α) = 0

— a stationary point is not enough.

Even vs odd. Even multiplicity touches without crossing; odd multiplicity

\geq 3

crosses with a horizontal inflection.

Multiplicity 1. A simple root is an ordinary crossing, with no flattening.

Losing a case. Solving

P^{'} (x) = 0

can give several candidates — test each in

P

Frequently asked questions

What is the multiplicity of a root?

It is how many times the factor $(x - α)$ appears in $P (x)$ .

How does calculus find a multiple root?

A root of multiplicity $m > 1$ satisfies both $P (α) = 0$ and $P^{'} (α) = 0$ , so solve them together.

What does the graph look like at a double root?

The curve is tangent to the $x$ -axis — it touches but does not cross.

What happens at a triple root?

The curve has a horizontal inflection: it crosses the axis while flattening.

Why is alpha a root of P prime?

By the product rule, $P^{'} (x) = (x - α)^{m - 1} [\dots]$ , so $α$ is a zero of $P^{'}$ of multiplicity $m - 1$ .

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Multiplicity of zeroes via calculus (P′ multiplicity, behaviour at roots)

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