# Polynomial Functions

## Overview

Welcome to the course material on Polynomial Functions in Further Mathematics. This topic delves into the realm of algebraic functions that play a fundamental role in mathematical modeling and problem-solving. By the end of this course, you will have a deep understanding of polynomial functions, their graphs, equations, and real-life applications.

One of the primary objectives of this course is to aid you in identifying and understanding polynomial functions. **Polynomial functions** are functions that can be expressed as an equation involving a sum of powers in one or more variables where the coefficients are constants. These functions play a crucial role in various branches of mathematics, physics, and engineering.

As we progress, you will learn how to recognize and sketch the graphs of polynomial functions. Graphical representation is a powerful tool in analyzing and interpreting functions. **Sketching graphs** allows us to visualize the behavior of functions, identify key characteristics such as roots and turning points, and comprehend the overall shape of the function.

Solving polynomial equations is another essential skill you will acquire through this course. Polynomial equations involve setting a polynomial expression equal to zero and determining the values of the variables that satisfy the equation. Through various methods such as factoring, synthetic division, and long division, you will master the art of **solving polynomial equations**.

The Fundamental Theorem of Algebra will also be explored in detail. This theorem states that every non-constant polynomial has at least one complex root. Understanding this theorem provides valuable insights into the **roots and factors** of polynomial functions, paving the way for deeper analytical approaches.

Furthermore, we will delve into the relationships between zeros, factors, and graphs of polynomial functions. By examining how the **zeros** of a polynomial function relate to its **factors** and **graph**, you will develop a comprehensive understanding of the interplay between these key components.

Real-life scenarios will be utilized to apply polynomial functions. From modeling growth patterns in populations to analyzing financial trends, **real-life applications** demonstrate how polynomial functions can be used to solve practical problems and make informed decisions.

Analyzing the behavior of polynomial functions at intercepts and end behavior will enhance your ability to interpret function graphs effectively. By studying how functions behave near intercepts and towards infinity, you will gain valuable insights into the **behavior of polynomial functions** in different contexts.

Transformations play a significant role in graphing polynomial functions. Understanding how **transformations** such as shifts, stretches, and reflections affect the graph of a polynomial function enables you to manipulate and visualize functions more efficiently.

The Division Algorithm for polynomials will be covered, along with the application of **synthetic division** and long division to divide polynomials. These methods provide systematic approaches to dividing polynomials and simplifying complex expressions, contributing to a more structured problem-solving process.

In conclusion, this course material on Polynomial Functions aims to equip you with a deep understanding of polynomial functions, their graphs, equations, and applications. By mastering the concepts and techniques introduced in this course, you will be well-prepared to tackle diverse mathematical challenges and appreciate the beauty of polynomial functions in the realm of mathematics.

## Objectives

1. Applying Synthetic Division and Long Division to Divide Polynomials
2. Applying Polynomial Functions in Real-Life Scenarios
3. Exploring the Relationships Between Zeros, Factors, and Graphs of Polynomial Functions
4. Analyzing the Behavior of Polynomial Functions at Intercepts and End Behavior
5. Recognizing and Sketching Graphs of Polynomial Functions
6. Utilizing Transformations to Graph Polynomial Functions
7. Identifying and Understanding Polynomial Functions
8. Understanding the Division Algorithm for Polynomials
9. Understanding the Fundamental Theorem of Algebra
10. Solving Polynomial Equations

## Lesson Note

Polynomial functions are a fundamental concept in mathematics, providing a bridge from arithmetic and algebra to higher-level topics such as calculus. These functions are composed of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents. Understanding polynomials is crucial for solving various mathematical problems and can also be applied to real-life scenarios such as engineering, physics, and economics.

## Lesson Evaluation

Congratulations on completing the lesson on Polynomial Functions. Now that youve explored the key concepts and ideas, its time to put your knowledge to the test. This section offers a variety of practice questions designed to reinforce your understanding and help you gauge your grasp of the material.

You will encounter a mix of question types, including multiple-choice questions, short answer questions, and essay questions. Each question is thoughtfully crafted to assess different aspects of your knowledge and critical thinking skills.

Use this evaluation section as an opportunity to reinforce your understanding of the topic and to identify any areas where you may need additional study. Don't be discouraged by any challenges you encounter; instead, view them as opportunities for growth and improvement.

1. Identify the degree of the polynomial function: f(x) = 3x^4 - 2x^2 + 5x - 1. A. 2 B. 3 C. 4 D. 5 Answer: C
2. What is the leading coefficient of the polynomial function: g(x) = -10x^3 + 2x^2 + x + 4? A. -10 B. 2 C. 1 D. 4 Answer: A
3. Given the polynomial function h(x) = x^5 - 3x^4 + 2x^3 + x - 7, what are the zeros of the function? A. -1, 1 B. 0, 2 C. -2, 3 D. 1, 7 Answer: C
4. If a polynomial function has a zero at x = -2, what is the factor of the polynomial? A. (x + 2) B. (x - 2) C. (x + 1) D. (x - 1) Answer: A
5. What is the end behavior of the polynomial function: p(x) = x^4 - 2x^3 + 3x^2 - x + 1? A. As x approaches negative infinity, p(x) approaches positive infinity. As x approaches positive infinity, p(x) approaches negative infinity. B. As x approaches negative infinity, p(x) approaches negative infinity. As x approaches positive infinity, p(x) approaches negative infinity. C. As x approaches negative infinity, p(x) approaches positive infinity. As x approaches positive infinity, p(x) approaches positive infinity. D. As x approaches negative infinity, p(x) approaches negative infinity. As x approaches positive infinity, p(x) approaches positive infinity. Answer: C
6. For the polynomial function q(x) = 2x^3 - 5x^2 + 3x + 1, what is the degree of the function? A. 2 B. 3 C. 4 D. 5 Answer: B
7. Which statement about the Fundamental Theorem of Algebra is true? A. A polynomial of degree n has exactly n zeros. B. Every polynomial function can be factored into linear and quadratic factors. C. The fundamental theorem states that every nth degree polynomial has n distinct roots. D. A polynomial of degree n can have at most n distinct real roots. Answer: C
8. What is the multiplicity of the zero x = 1 for the polynomial function: r(x) = (x - 1)^3 (x + 2)? A. 1 B. 2 C. 3 D. 4 Answer: C
9. Which of the following is a possible equation for the graph shown: y = (x + 2)(x - 1)(x - 3)? A. y = x^3 + 2x^2 - 3x - 6 B. y = x^3 - 2x^2 - 3x + 6 C. y = x^3 - 6x^2 + 11x - 6 D. y = x^3 + 10x^2 - 11x - 6 Answer: A

## Past Questions

Wondering what past questions for this topic looks like? Here are a number of questions about Polynomial Functions from previous years

Question 1

Two functions f and g are defined on the set of real numbers, R, by

f:x → x2  + 2 and g:x → 1x+2.Find the domain of (g∘f)1

Practice a number of Polynomial Functions past questions