Topic One (1.1): Algebra

Introduction

The Modulus Function: The modulus of a number is the magnitude of that number without a sign attached. It is also known as the absolute value of the number. For example, the modulus of LaTeX: -1 $-1$ written as LaTeX: |-1| $|-1|$ is LaTeX: 1 $1$ . The modulus of LaTeX: x $x$ , written LaTeX: |x| $|x|$ is $x$ for values of $LaTeX: x\ge0$ $x\ge0$ and LaTeX: -x $-x$ for values of LaTeX: x<0 $x<0$ . So, the graph of LaTeX: y=|x| $y=|x|$ is LaTeX: y=x $y=x$ , for all positive values of $x$ and LaTeX: y=-x $y=-x$ for all negative values of $x$ .

A polynomial is a function of the form LaTeX: ax^n $ax^n$ where $a$ is a constant (this means that it has a fixed value) and LaTeX: n $n$ is a positive integer. The 'degree of order' of the polynomial is the highest power of LaTeX: x $x$ . The division algorithm for polynomials is: dividend $=$ divisor $LaTeX: \times$ $\times$ quotient $+$ remainder.

The Remainder Theorem is a nice simple method that helps you find the remainder when a polynomial is divided by a linear function. The theorem states that when $LaTeX: f\left(x\right)$ $f\left(x\right)$ is divided by $LaTeX: \left(x-a\right)$ $\left(x-a\right)$ the remainder is $LaTeX: f\left(a\right)$ $f\left(a\right)$ . This means that when you are given the equations to be divided, then the remainder is the value of the equation when LaTeX: x=a $x=a$ .

The Factor Theorem is linked to the remainder theorem in that if $LaTeX: \left(x-a\right)$ $\left(x-a\right)$ is a factor of the polynomial, there will be no remainder. So, $LaTeX: f\left(a\right)=0$ $f\left(a\right)=0$ . Therefore, if you want to find out if $LaTeX: \left(x-a\right)$ $\left(x-a\right)$ is a factor of $LaTeX: f\left(x\right)$ $f\left(x\right)$ , just check that $LaTeX: f\left(a\right)=0$ $f\left(a\right)=0$ . The factor theorem can also be used to factorise polynomials of greater degree than LaTeX: 2 $2$ and therefore helps us solve some cubic, quartic, etc. equations.

Reading

For this topic, you will need to work through pages 1 to 24 of your textbook.

Tasks

Complete the following:

Start by reading through pages 1 to 2 as an introduction to this chapter.

The modulus function

Work through pages 2 to 6, including exercise 1A.
There is a brilliant set of video tutorials on the modulus function, which starts at:

WATCH – The Modulus Function (Exam Solutions) Links to an external site.

Graphs of $LaTeX: y=|f\left(x\right)|$ $y=|f\left(x\right)|$ where $LaTeX: f\left(x\right)$ $f\left(x\right)$ is linear

Work through pages 7 to 8, including exercise 1B.
If you need extra help, try watching:

WATCH – Graphing $LaTeX: y=|f\left(x\right)|$ $y=|f\left(x\right)|$ (Exam Solutions)

Solving modulus inequalities

Work through pages 8 to 11, including exercise 1C.
If you need more help, watch these videos:

WATCH – Modulus Equations (Exam Solutions) Links to an external site.

WATCH – Modulus Inequalities (Exam Solutions) Links to an external site.

Division of polynomials

Work through pages 11 to 14, including exercise 1D.
This video explains algebraic division:

WATCH – Algebraic Long Division (Exam Solutions) Links to an external site.

The factor theorem

Work through pages 14 to 18, including exercise 1E.

The remainder theorem

Work through pages 18 to 21, including exercise 1F.

WATCH – Introduction to the Factor and Remainder Theorems (YouTube) Links to an external site.

WATCH – Basic Examples of the Factor and Remainder Theorems (YouTube) Links to an external site.

WATCH – Solving Cubic Equations Using the Factor Theorem (YouTube) Links to an external site.

Consolidate

Now consolidate your understanding by making revision cards for this chapter (the checklist on page 21 will help) and work through the end-of-chapter review on pages 22 to 24.