Learning Resources

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Outcomes

By the end of this lesson you will be able to answer questions like the ones in the pre-test below. 

Approach #1

Answer the questions on the pre-test below in your Math 3103 binder. Then refer to the pages of the lesson that contain instruction on the questions you were unable to do, for which you got an incorrect answer, or for which you would like some further explanation.

Approach #2

Proceed directly to the first page of the lesson without attempting the pre-test. To do this simply click on the Lesson button above. You can then return to this pre-test later and use it as a worksheet.

Pre-test

1. Given the polynomial 4x3 - 5x2 - x4 + 1 , identify the following:
    (a) the type of polynomial (i.e. linear, quadratic, etc.)
    (b) the leading coefficient.
    (c) the constant term.
    (d) the linear term.

2. Find the exact root of each of the following:
    (a)   x - 5 = 10                 (b)  3x - 3 = 0                          (c) 

    (d)  3x + 5 = 2x - 2          (e)  2(x - 1) = 10                     (f)  

    (g)       (h)

    (i)  

3. Find the exact zero of each of the following linear functions:

    (a)  f(x) = 2x - 4                      (b)                  (c) 

4. The exact x and y intercepts of the graphs of the following:

    (a)  3x + 2y = 6                      (b)  

Answers

Students will be expected to:

  • Recognize and use the language of polynomials.
  • Solve linear equations.
  • Factor polynomial expressions.
  • Use factoring to solve polynomial equations in one variable of degree four or less.
  • Use the rational roots theorem to help solve polynomial equations in one variable of degree four or less.
  • Use synthetic division in connection with the rational roots theorem to obtain a depressed equation.

Introduction

This section consists of four lessons as follows:

Lesson 01 Linear Equations
In this lesson you will review the concept of, and terminology associated with, polynomials. In particular you will review linear polynomials and review how to find the root of an equation which contains them.

Lesson 02 Factoring Polynomials
This lesson reviews some of the techniques of factoring polynomials. In it you will review common factors, difference of squares, factoring trinomials of the form ax2 + bx + c , and grouping.

Lesson 03 Solving Polynomial Equations Using Factoring
In this lesson you will apply the techniques of factoring to find the roots of polynomial equations of degree four or less.

Lesson 04 The Rational Roots Theorem
Sometimes the factors of a polynomial may not be obvious and we need to turn to other methods. This theorem is one of those methods. It relates the roots of any polynomial equation to its leading coefficient and constant term. Using this relation you will test to see if the given equation has any rational roots and if it does, use synthetic division to depress the equation to a lower order.

Prerequisites

To be successful in this lesson, it would be helpful to know the following:

  • Operations of addition, subtraction, multiplication and division with rational numbers and radicals.
  • Terminology associated with polynomials.
  • Concept of root of an equation and zero of a function.
  • Concept of the x and y intercepts of a graph.