Higher Maths · Worksheet 5

Quadratics and Polynomials Worksheet | Higher Maths

Skills covered: Factorising; quadratic roots and the expression b-squared minus four ac; tangency; factor and remainder theorems; cubic intersections.

18 questions68 marks3 graded sectionsFree PDF

Download PDF Polynomials revision notes

Instructions: Attempt all questions. Show sufficient working to justify your answers. Give exact values unless a degree of accuracy is stated. Each section is separate, so you can target the part you need.

These are the same worksheets our pupils use at the Pollokshields tuition centre. Start with Section A if quadratics and polynomials still feels shaky; if you are confident, go straight to Section C and time yourself.

Section A

Prerequisite core skills

Revise earlier learning

Question 1

[2 marks]

Factorise fully

3x^2 - 11x - 4

.

Question 2

[3 marks]

Solve

2x^2 + 3x - 9 = 0

.

Question 3

[3 marks]

Determine the nature of the roots of

3x^2 - 6x + 5 = 0

. Give a reason.

Question 4

[3 marks]

Find

k

if

x^2 + 6x + k = 0

has equal roots.

Question 5

[2 marks]

Find the remainder when

P(x) = x^3 - 2x^2 - 5x + 6

is divided by

x - 2

.

Question 6

[3 marks]

Factorise

x^3 - 4x

fully and solve

x^3 - 4x = 0

.

Section B

Topic practice

Build individual techniques

Question 7

[5 marks]

Show that

x = -1

is a root of

x^3 - 4x^2 + x + 6 = 0

. Hence solve the equation.

Question 8

[4 marks]

P(x) = x^3 + ax^2 - 7x + 5

has a factor

x - 1

. Find

a

and the remainder when

P(x)

is divided by

x + 2

.

Question 9

[5 marks]

The line

y = 2x + k

is a tangent to

y = x^2 - 4x + 7

. Find

k

and the point of contact.

Question 10

[4 marks]

Find all real values of

p

for which

x^2 + (p + 1)x + 2p = 0

has equal roots.

Question 11

[4 marks]

Factorise

x^3 - 3x^2 - 4x + 12

and state the

x

-intercepts of the corresponding cubic.

Question 12

[4 marks]

Find the coordinates of the points of intersection of

y = x^2 - 3x + 1

and

y = x + 1

.

Question 13

[4 marks]

Find all the points where

y = 2x^3 - 5x^2 - 3x

meets the

x

-axis.

Question 14

[3 marks]

The polynomial

P(x) = x^3 - 2x^2 + mx + 6

has a factor

x + 1

. Find

m

.

Section C

Exam-style and challenge

Combine skills, exam-style

Question 15

[5 marks]

Show that

x = 1

is a root of

x^2 + (k + 2)x - k - 3 = 0

for every real

k

. Find

k

if the roots are equal.

Question 16

[5 marks]

Find the possible values of

m

for which

y = mx + 2

is tangent to

y = x^2 - 4x + 6

.

Question 17

[4 marks]

Solve the equation

x^4 - 5x^2 + 4 = 0

.

Question 18

[5 marks]

P(x) = x^3 + kx^2 - 4x - 4k

. Factorise

P(x)

fully and state the values of

k

for which

P(x) = 0

has three distinct real roots.

Looking for the answers?

We do not publish the worked answers online. If you get stuck, bring the worksheet to a session and one of our Higher Maths tutors will go through the method with you, line by line, the way the SQA marking instructions award it.

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Frequently asked questions

Are the answers to these worksheets available online?

No. We keep the worked answers off the website on purpose. Pupils learn far more by attempting every question, checking against their notes and then going over the method with a tutor. Our students work through the full worked solutions in class, and if you are stuck on a particular question you can book a session and we will go through it with you step by step.

Are these official SQA or Qualifications Scotland papers?

No. Every question is newly written by Excel in MathSci to match the Higher Maths course specification. They are original practice resources, not official Qualifications Scotland papers, so use them alongside the real past papers rather than instead of them.

How should I use the three sections?

Section A checks the prerequisite skills you need before the topic makes sense, so start there if the topic feels shaky. Section B builds each technique on its own. Section C is exam-style and combines several skills, so treat it as a timed test once Sections A and B feel comfortable.

Can I print the worksheet?

Yes. Download the free PDF, which has space to write your working under every question, or print this page directly. The worksheet is free for personal study; please ask before reproducing it for a class or a commercial service.

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