BBE Mathematics Unit 1 - Chapter 3 Notes (DU - Maharaja Agrasen College): Polynomials, Power & Exponential Functions

Mathematics is a core subject in the Bachelor of Business Economics (BBE) curriculum. A strong understanding of functions and their properties forms the foundation for economics, statistics, finance, and business analytics. Unit 1, Chapter 3 of BBE Mathematics introduces students to polynomial functions, power functions, exponential functions, quadratic functions, and the remainder theorem—all essential tools for analyzing quantitative relationships in business and economics.

This comprehensive 1200-word guide explains each topic in a clear, exam-oriented way, helping students revise effectively and strengthen conceptual understanding.

1. Polynomial Functions

Polynomial functions are fundamental to understanding relationships between variables. They are used in economics to model revenue, cost, and profit functions.

Definition

A polynomial function is a function of the form:

P(x) = a₀ + a₁x + a₂x² + ... + anxⁿ

Where:

• a₀, a₁, …, an are constants (coefficients)

• n is a non-negative integer, known as the degree of the polynomial

Types of Polynomial Functions

1. Linear Polynomial – Degree 1 → P(x) = ax + b

2. Quadratic Polynomial – Degree 2 → P(x) = ax² + bx + c

3. Cubic Polynomial – Degree 3 → P(x) = ax³ + bx² + cx + d

4. Quartic & Higher Degree Polynomials → Degree ≥ 4

Key Features

• Degree determines the shape of the graph

• Leading coefficient affects the end behaviour of the graph

• Polynomials are continuous and differentiable

Applications in Business & Economics

• Cost functions: C(x) = ax² + bx + c

• Revenue functions: R(x) = px – qx²

• Profit functions: π(x) = R(x) – C(x)

2. Quadratic Functions

A quadratic function is a specific type of polynomial function of degree 2.

General Form

f(x) = ax² + bx + c, a ≠ 0

Key Properties

• Vertex: Turning point of the parabola

• Axis of symmetry: x = –b / 2a

• Direction: Opens upwards if a > 0; downwards if a < 0

• Roots/Zeros: Solve ax² + bx + c = 0 using factorization, completing square, or quadratic formula

Graph

• Parabolic curve

• Minimum or maximum at the vertex

• Intercepts with x-axis (real roots) and y-axis (constant term c)

Economic Applications

• Maximizing profit or utility functions

• Cost and revenue analysis with quadratic behaviour

• Modeling diminishing returns

3. Power Functions

Power functions are widely used to model growth, elasticity, and scale effects.

Definition

A power function is a function of the form:

f(x) = k · x^n

Where:

• k is a constant

• n is the exponent (positive, negative, or fractional)

Types of Power Functions

1. Direct Proportionality (n = 1): Linear growth → f(x) = kx

2. Quadratic Growth (n = 2): f(x) = kx²

3. Cubic and Higher Growth (n > 2): Accelerating growth

4. Fractional/Negative Powers: Represent diminishing returns or decay

Applications in Economics

• Production functions (Cobb-Douglas type)

• Economies of scale

• Demand elasticity analysis

4. Exponential Functions

Exponential functions are important for modeling growth and decay in business and economics.

Definition

An exponential function has the form:

f(x) = a · b^x, a ≠ 0, b > 0, b ≠ 1

Where:

• a is the initial value

• b is the base, representing growth (>1) or decay (<1)

Graph Characteristics

• Continuous and smooth

• Rapid increase if b > 1 (growth)

• Rapid decrease if 0 < b < 1 (decay)

• Never touches x-axis (asymptotic behaviour)

Applications

• Compound interest

• Population growth

• Depreciation of assets

• Exponential decay in inventory or pricing models

5. Remainder Theorem

The remainder theorem is a useful tool for dividing polynomials quickly. It simplifies polynomial calculations without full division.

Statement

If a polynomial P(x) is divided by (x – a), the remainder is P(a).

Formula

P(x) ÷ (x – a) = Q(x) + R
Where R = P(a)

Example

If P(x) = x³ – 4x² + 5x – 2, divide by (x – 1):

• Remainder = P(1) = 1 – 4 + 5 – 2 = 0

• Hence, (x – 1) is a factor

Applications

• Quick factorization

• Root finding in polynomial equations

• Simplifying economic models using polynomials

6. Summary of Functions and Their Graphs

Graphical understanding complements algebraic knowledge. By plotting these functions, students can visualize behaviour and trends in business and economics.

7. Importance for BBE Students

Understanding Chapter 3 is essential for:

• Economics: Analyzing cost, revenue, profit, and demand

• Finance: Compound interest, depreciation, growth models

• Business Analytics: Trend modelling using polynomial and exponential functions

• Quantitative Methods: Preparing for statistics and econometrics

These concepts form the building blocks for future quantitative courses in BBE, enabling students to translate mathematical theory into real-world business applications.

Conclusion

Chapter 3 of BBE Mathematics (Unit 1) introduces students to key functional forms and polynomial tools that are widely used in economics and business. From polynomials and quadratics to power and exponential functions, and the remainder theorem, mastering these concepts strengthens analytical thinking and problem-solving skills.

With this chapter’s notes, DU BBE students from Maharaja Agrasen College can:

• Visualize and graph functions

• Apply polynomial and exponential models to business problems

• Solve complex equations efficiently using the remainder theorem

By combining conceptual clarity, graphical understanding, and practical applications, these notes provide a strong foundation for success in BBE mathematics and related subjects.