Binomial Theorem - Mathematics (Undergraduate Foundation)

Stop manually multiplying brackets. This course teaches the systematic expansion of binomial expressions using Pascal's triangle and the formal Binomial Theorem. We cover the fundamental proof for positive integers before moving to individual term identification and the general expansion for negative or fractional powers. You will master the mechanics of series expansion from simple squares to complex infinite series. The Binomial Theorem is a core tool in probability, statistics, and financial engineering. It allows for the approximation of complex functions and the calculation of compounding interest or risk factors. Engineers and data scientists use these principles to manage error margins and optimise algorithms. Understanding these patterns is essential for any career involving quantitative analysis or technical forecasting. You will expand binomials to any power and identify specific terms within a series without full expansion. You will demonstrate the theorem's proof and apply Pascal's triangle for rapid computation. You will also calculate expansions for non-integer indices and determine the range of validity for these series. These skills enable the simplification of advanced algebraic expressions. This curriculum is for undergraduate students and secondary school leavers entering science or technology tracks. It provides the necessary foundation for engineering, physics, and economics degrees. Professionals needing a mathematical refresher for data analysis will find the direct approach efficient. Any learner requiring precise algebraic tools for higher education will gain immediate value.

6 hrs

Enrolment valid for 12 months

This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.

MTH 101: Elementary Mathematics I - Algebra and Trigonometry

Master the foundational mathematical structures essential for success in quantitative undergraduate degrees and professional technical roles. This comprehensive learning track delivers a rigorous treatment of algebra and trigonometry, moving rapidly from fundamental set theory and real number operations to advanced topics including matrix algebra, complex numbers, and analytical trigonometry. You will establish the critical problem-solving framework required for advanced study in calculus, engineering mechanics, and data science. This programme is primarily designed for first-year university students in STEM disciplines requiring strong analytical bases, particularly engineering, physics, computer science, and economics. It also serves as an intensive, high-level refresher for professionals returning to academia or shifting into data-driven roles demanding precise numerical literacy and logical structuring. Prior competence in standard secondary school mathematics is assumed; focus is placed strictly on mastery and application of core definitions. Upon completion, you will possess the skills to construct rigorous logical arguments using set theory and mathematical induction, model complex relationships with functions and matrices, and analyze periodic systems using advanced trigonometry. You will demonstrate competence in solving diverse equation types, from quadratics to linear systems, and manipulating complex numbers in engineering applications. This track prepares you directly for the mathematical demands of second-year university studies and technical professional certification exams.

Course Chapters

1. Introduction

This chapter covers basic binomial expansion and Pascal's triangle. Identifying these patterns early allows you to skip long multiplication and build algebraic speed. Accuracy in these fundamentals is required for all advanced scientific computations. You will learn to find the total number of terms; use Pascal???s triangle for coefficients; observe the symmetry of powers; and expand simple binomials to small positive indices.

Concept Overviews

3 Lessons

36:37

Problem Walkthroughs

1 Lesson

7:03

2. The Binomial Formula

This chapter covers the formal binomial formula, combination notation, and rigorous proof by mathematical induction. These tools allow you to find specific terms and coefficients for large powers without full expansion. Mastering these techniques is essential for accurate decimal approximations and university-level algebraic proofs. You will master the nCr formula for coefficients; calculate the general term of an expansion; identify terms independent of x; prove the theorem using mathematical induction; and solve small decimal approximations.

Concept Overviews

2 Lessons

1:22:03

Problem Walkthroughs

2 Lessons

16:38

3. Individual Terms

This chapter focuses on isolating specific parts of a binomial expansion without writing out the full polynomial. It is a critical skill for solving complex engineering and probability problems where only a single value or coefficient is required. By the end, you will master calculating the general term formula; finding the coefficient of a specific power of x; and identifying the term independent of x. You will also learn to determine middle terms and specific term positions.

Concept Overviews

1 Lesson

16:31

Problem Walkthroughs

6 Lessons

39:25

4. Negative and Fractional Powers

This chapter extends binomial expansion to expressions with negative and fractional indices. These expansions result in infinite series rather than finite terms, providing a critical tool for approximating roots and reciprocal functions. Engineers and scientists use these series to simplify complex equations where absolute precision is required for very small values. You will master the general expansion formula for any rational index; determine the range of values for which an expansion is valid; calculate approximations for square and cube roots; and find the first few terms of infinite binomial series.

Concept Overviews

1 Lesson

34:22

Problem Walkthroughs

4 Lessons

34:47