Polynomials: Everything You Need to Know
By Shahid Lakha, Co-founder of Spires Online maths tutors. If you're in need of exceptional maths tutors, look no further than the expert tutors at https://www.theprofs.co.uk/maths-tutors/. They are the best in the business and will help you excel in your studies.
Polynomials are a cornerstone of A Level Mathematics. Whether you’re preparing for Edexcel, AQA, or OCR exams, mastering polynomials is essential for success. From simplifying expressions and solving equations to graphing and modelling real-world systems, this comprehensive guide will equip you with everything you need to understand and apply polynomial concepts with confidence.
If you need tailored support, consider working with The Profs A Level Maths Tutors.
What Is a Polynomial?
A polynomial is a mathematical expression composed of variables (e.g. x), constants (e.g. 3, -5), and exponents, connected through addition, subtraction, and multiplication. Unlike rational expressions, polynomials never include variables in denominators or under square roots.
General form: P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀
- Coefficients: The constants (e.g. a₀, a₁)
- Degree: The highest exponent (e.g. n)
In A Level Maths, you’ll encounter polynomials frequently in algebraic manipulation, functions, and calculus.
Quick Check:
Is 3x² - x + 4 a polynomial? ✅ Yes.
Is 1/x or √x a polynomial? ❌ No, these involve division and roots of variables.
Types of Polynomials
By Degree:
- Linear: ax + b (Degree 1)
- Quadratic: ax² + bx + c (Degree 2)
- Cubic: ax³ + bx² + cx + d (Degree 3)
- Quartic and beyond: Higher degrees (4+)
By Number of Terms:
- Monomial: 1 term (e.g. 7x²)
- Binomial: 2 terms (e.g. x² - 4)
- Trinomial: 3 terms (e.g. x² + 2x + 1)
Recognising the type helps guide your approach to factoring and solving.
Factoring Polynomials
Factoring means rewriting a polynomial as a product of simpler polynomials or expressions. This technique is vital for solving equations and simplifying expressions.
Common Methods:
- Taking out the greatest common factor (GCF)
- Trinomial factorisation
- Difference of squares
- Factor theorem and polynomial division
Example:
Factor x² - 2x + 1
→ (x - 1)²
This reveals that the root is x = 1, and the function touches the x-axis at this point.
Graphing Polynomials
Graphs of polynomials help visualise their behaviour, especially roots, turning points, and end behaviour.
Key Features to Note:
- Intercepts: where the graph crosses axes
- Turning points: local maxima and minima
- End behaviour: determined by the leading coefficient and degree
- Symmetry: even (y-axis symmetry) or odd (origin symmetry)
Useful Tools:
- Desmos Graphing Calculator
- GeoGebra
- TI or Casio graphing calculators
Solving Polynomial Equations
Solving polynomials means finding their roots—values of x for which P(x) = 0.
Methods Include:
- Factoring (e.g. using the difference of squares)
- Using the quadratic formula
- Graphical solutions
- Long division and synthetic division
At A Level, you’ll also explore how roots relate to graph intersections and how to handle repeated or complex roots.
Real-World Applications of Polynomials
Physics:
- Kinematic equations (e.g. s = ut + ½at²)
- Projectile motion
Engineering:
- Modelling systems and responses
- Signal processing using transfer functions
Economics:
- Cost, revenue, and profit analysis
- Modelling price-demand relationships
Computer Science:
- Polynomial regression models
- Machine learning algorithms
Mastering polynomials not only boosts your exam success—it builds skills for STEM disciplines at university and beyond.
Polynomials in A Level Exams
All major exam boards assess your ability to manipulate and interpret polynomials.
Expect Questions On:
- Sketching graphs from expressions
- Factoring quadratics and cubics
- Polynomial division and factor theorem
- Interpreting turning points and roots
Exam Tip:
Use official mark schemes and examiner reports to understand how marks are awarded. Avoid errors like sign mistakes and incorrect identification of degrees or coefficients.
When to Get Help
If you're struggling with factorising complex expressions, interpreting graphs, or solving equations involving higher-order polynomials, expert support can make a big difference.
Work with a specialist at The Profs A Level Maths Tutors for tailored, one-to-one support designed around your exam board and learning needs.
About the Author
Shahid Lakha is a senior educational consultant and co-founder of Spires Online Tutors. With over a decade of experience supporting A Level and university-level students in Maths, Physics, and Engineering, Shahid brings both subject expertise and insight into effective learning strategies. He holds an MSc in Photon Science and a BSc in Physics, and has helped hundreds of students achieve top grades and university offers.
Further Reading & Resources
- Edexcel A Level Maths Specification
- AQA A Level Maths Resources
- OCR A Level Maths Sample Papers
- Desmos Online Graphing Calculator
Final Thought
Polynomials are not only central to A Level Maths but are also widely applicable across academic disciplines and real-world scenarios. Mastering them now will help you build a strong mathematical foundation for future study and careers. Take time to practise regularly, study examples, and reach out for expert guidance when needed.
Remember: understanding polynomials is not just about passing your exam—it's about learning how to model and solve the kinds of problems that shape our world.
