Higher Maths
Polynomials and Quadratics

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Course content

  • All Nat 5 work on quadratics, linear inequalities and completing the square is assumed.
  • Factorising a cubic or quartic polynomial expression
  • Solving a cubic or quartic polynomial equation
  • Using the discriminant to find an unknown, given the nature of the roots of an equation
  • Solving quadratic inequalities, ax2+bx+c0 (or 0)
  • Completing the square in a quadratic expression where the coefficient of x2 is non-unitary
  • Finding the coordinates of the point(s) of intersection of a straight line and a curve or of two curves.

Textbook page references

  • Zeta Higher Mathematics pp.23-43 and 101-111
  • Heinemann Higher Maths pp.131-163
  • TeeJay Higher Maths pp.60-67 and 108-119
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Discriminant

For a quadratic expression ax2+bx+c, the discriminant is defined as b24ac.

The discriminant helps us discriminate between different types of quadratic expression.

If b24ac<0, the expression has no real roots.

If b24ac=0, the expression has two equal real roots (a repeated root).

If b24ac>0, the expression has two distinct real roots.

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Unitary

A unitary quadratic has 1 as the coefficient of x2.

Examples:
x27x+3
x25
3+8x+x2

Non-unitary

In a non-unitary quadratic, the coefficient of x2 is not equal to 1.

Examples:
2x2x+1
x23
7x3x2

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Example 1 (non-calculator)

(a)  Show that 2x+3 is a factor of 2x3+3x22x3.
(b)  Hence factorise 2x3+3x22x3 fully.

Example 2 (non-calculator)

Factorise 2x3+9x26x5 fully.

Example 3 (non-calculator)

(a)  Show that x+2 is a factor of 2x4+9x3x218x+8.
(b)  Hence factorise 2x4+9x3x218x+8 fully.

Recommended textbook

Zeta Maths: Higher Mathematics 
 Best price, direct from the publisher

Example 4 (non-calculator)

(a)  Show that x3 is a factor of 2x43x319x24.
(b)  Hence factorise 2x43x319x24 fully.

Example 5 (non-calculator)

For the polynomial x3x2+mx+n

  • x3 is a factor
  • 54 is the remainder when it is divided by x5

(a)  Determine the values of m and n.
(b)  Hence solve x3x2+mx+n=0.

Example 6 (non-calculator)

The same remainder is found when x36x2+2xp and x3+5x2+(2p+1)x37 are divided by (x+2). Find the value of p.

Revision guides

How to Pass Higher Maths 
BrightRED Higher Maths Study Guide 

Example 7 (non-calculator)

Solve x3+3x24x12=0.

Example 8 (non-calculator)

Solve x4x310x2+4x+24=0.

Example 9 (non-calculator)

The graph of y=f(x), where f(x)=k(xa)(xb)2, has a minimum turning point at (3,0), a root 2 and passes through the point (1,48). Find the values of a, b and k.

Practice papers

Essential Higher Maths Exam Practice 
Higher Practice Papers: Non-Calculator 
Higher Practice Papers: Calculator 

Example 10 (non-calculator)

Find the values of k for which x2+(k+3)x+4=0 has equal roots.

Example 11 (non-calculator)

Find the range of values of p for which 2x2+5x+p+1=0 has no real roots.

Example 12 (non-calculator)

Find the range of values of a for which x26x+a=0 has two distinct real roots.

Stationery supplies

Pukka Pad: A4 squared notepads 
Uni-ball Eye: fine-tip rollerball pens 

Example 13 (non-calculator)

Find the range of values of n for which x2nx+3n=0 has two distinct real roots.

Example 14 (non-calculator)

A rectangle has length x cm and a breadth that is 1 cm shorter than the length. Its area is less than 30 cm2. Find the range of possible values of x.

Example 15 (non-calculator)

Express 2x2+12x+5 in the form a(x+b)2+c.

Scientific calculators

Casio FX-85GTCW scientific calculator 
Casio FX-991CW advanced calculator 

Example 16 (non-calculator)

Express 4x228x1 in the form p(x+q)2+r.

Example 17 (non-calculator)

Determine the point(s) of intersection of the parabola y=x2+3x7 and the line y=4x1.

Example 18 (non-calculator)

The line y=5x3 and the curve y=x38x+9 intersect at three points. One of these points is (3,12). Find the coordinates of the other two points of intersection.

Books for teachers

Jo Boaler: Mathematical Mindsets 
Craig Barton: Tips for Teachers 

Example 19 (non-calculator)

SQA Higher Maths 2023 Paper 1 Q5

The equation 2x2+(3p2)x+p=0 has equal roots. Determine the possible values of p.

Example 20 (non-calculator)

SQA Higher Maths 2023 Paper 1 Q10

(a)  Show that (x+5) is a factor of x4+3x37x2+9x30.
(b)  Hence, or otherwise, solve x4+3x37x2+9x30=0, xR.

Example 21 (non-calculator)

SQA Higher Maths 2024 Paper 1 Q8

The equation x2+(m4)x+(2m3)=0 has no real roots. Determine the range of values of m. Justify your answer.

Example 22 (non-calculator)

SQA Higher Maths 2024 Paper 1 Q10

(a)  Show that (x1) is a factor of 2x4+3x34x23x+2.
(b)  Hence, or otherwise, factorise 2x4+3x34x23x+2 fully.

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Past paper questions

Factors and remainders:
Specimen Paper 1 Q8
2015 Paper 1 Q3
2016 P2 Q3 (with integration)
2017 Paper 2 Q2
2018 Paper 2 Q7 (with sequences)
2019 Paper 2 Q10
2021 Paper 1 Q10
2022 Paper 1 Q13
2023 Paper 1 Q10
Identify coefficients of a cubic:
Specimen Paper 1 Q9
Identify a polynomial, given its roots:
2016 Paper 1 Q15
2018 P1 Q15 (with differentiation)
2018 Paper 2 Q3
Intersection of polynomial and line:
Spec. P2 Q6 (with differentiation)
2018 Paper 1 Q7
Completing the square:
2015 Paper 2 Q2 (with functions)
2016 Paper 1 Q12 (with functions)
2017 Paper 2 Q4(a)
2018 Paper 2 Q4
2019 P2 Q7(a)
2022 Paper 1 Q11
2023 Paper 1 Q12
Discriminant:
Specimen Paper 1 Q6
2016 Paper 2 Q2
2017 Paper 1 Q4
2018 Paper 2 Q10
2019 Paper 1 Q2
2021 Paper 1 Q1
2022 Paper 2 Q2
2023 Paper 1 Q5
Quadratic inequalities:
2015 Paper 1 Q8
2017 Paper 2 Q8 (with sequences)
2018 Paper 2 Q10
2022 Paper 2 Q5(b) (with functions)

Other great resources

Detailed notes - HSN
Detailed notes - Rothesay Academy
Revision notes - BBC Bitesize
1. Dividing and factorising
2. Solving polynomial equations
Notes - Airdrie Academy
1. Quadratics
2. Polynomials
Notes and examples - Maths Mutt
Key points - Perth Academy
1. Quadratic functions
2. Polynomials
Notes and videos - Mistercorzi
1. Quadratic theory revisited I
2. Quadratic theory revisited II
3. Polynomials and synthetic division
4. Factors, roots and graphs
Lesson notes - Maths 777
1. Evaluation; nested form
2. Remainder; synthetic division
3. Factor theorem
4. Polynomial roots
5. Sketching polynomials
6. Polynomial functions from graphs
Videos - Larbert High School
• Polynomials:
1. Introduction
2. Dividing polynomials
3. Factorising polynomials
4. Finding unknown coefficients
5. Solving equations
6. Finding functions from graphs
• Quadratics:
1. Completing the square
2. Inequations
3. Discriminant
4. Using the discriminant
5. Intersecting parabolas and lines
Videos - Maths180.com
Videos - Mr Thomas Maths
1. Polynomials
2. Quadratics
Videos - Siōbhán McKenna
1. Polynomials
2. Quadratics
Resources - MathsRevision.com
PowerPoint
Mindmap
Practice questions
Worksheets - Brannock High School
1. Quadratic inequalities (Answers)
2. Completing the square (Answers)
3. Synthetic division (Answers)
4. Discriminant (Answers)

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