Lesson 1.3 Division of polynomials

This is a free lesson. We trust you enjoy it!

Note: this is a fairly long lesson, so you may want to take it over two days — depending, of course, on how you have worked out your schedule.

The concept of dividing polynomials by each other

Division is, of course, one of the first things you learned when you started learning arithmetic. No doubt you remember being asked, "If you have nine sweets and three children, how many sweets does each child get?" Dividing polynomials is a little more complicated than \( 9 \div 3 \), but we will see that we define the division of polynomials by following the pattern of dividing integers (with which you are very familiar).

The first video explains the idea of dividing polynomials by each other, introduces important terms like divisor, quotient, and remainder, and provides definitions of these terms.

For your notebook:

Division of polynomials

When a polynomial \( p(x) \) is divided by another polynomial \( d(x) \) (with degree greater than zero), \( d(x) \) is known as the divisor, and the quotient \( q(x) \) and remainder \( r(x) \) are defined by the identity

 $$ p(x) \equiv d(x)q(x) + r(x) $$

where the degree of the remainder is less than the degree of the divisor.

The degree of the quotient added to the degree of the divisor is equal to the degree of \( p(x) \).

 Examples of dividing polynomials

We will now work through two examples of polynomial division. In each case we will find the quotient and remainder when one polynomial is divided by another.

As we work through these examples, we will learn two methods of dividing polynomials. The first is the method of equating coefficients (which you practised in the last lesson), and the second is the method of long division. No doubt you learned long division some years ago (when working with integers and decimal numbers), and you may be surprised to know that it can be used for dividing polynomials. You will soon see, however, that it provides quite a handy and useful way of dividing polynomials, once you learn the technique.

Example 1

\( (x^2+2x+3) \div (x-1) \)

The next two videos demonstrate the principles for a fairly simple polynomial division. We start with the method of equating coefficients.

 And now for long division. This video explains the essence of the method.

Example 2

\( (3x^4 + 7x^3 + 10x^2 + 3) \div (3x^2 - 2x + 1) \)

This example is a bit more complex than the previous one. The methodologies are exactly the same, but the quantity of detail means that it easy to make mistakes. Take careful note of the comments in the videos which explain how to avoid simple, unnecessary errors.

The explanation of long division in this video points out one or two important issues, so watch carefully and take note!

The remainder theorem

By looking carefully at the definition of polynomial division (see note box above), we can develop a very useful theorem that applies when the divisor polynomial is of first order (i.e. it is linear). This theorem is known as the remainder theorem.

The next video explains the remainder theorem.

In the last video we dealt with a linear divisor of the form \( x - t \). But what do we do if we have a divisor like \( 5x-3 \)? Can we extend the remainder theorem to cover this situation? The next video shows how.

For your notebook:

The remainder theorem

When a polynomial \( p(x) \) is divided by \( (x-t) \), the remainder is the constant \( p(t) \).

When a polynomial \( p(x) \) is divided by \( (sx-t) \), the remainder is the constant \( p(  { t \over s } ) \). (This may be termed the 'extended form' of the remainder theorem.)

 Integrated problem

We conclude with a more integrated problem involving the remainder theorem. This is Example 1.4.4 from your text book.

Read section 1.4 in your text book to recap the explanations in the videos.

Weight lifting

Complete the following exercises from your text book:

Exercise 1C

  • 1 a, c, e
  • 2 a, c, e
  • 3 a, c
  • 4 a, c, e, g
  • 5, 7, 9, 11

If you are registered for this course, you can download the answer key from your download area on the My Account menu.