Polynomials Part1

Contents 1 Notes 1.1 Learning Objectives (Syllabus) 1.2 Prior Knowledge 2 Degree & Coefficients 2.1 Example 2.2 Common Names 2.3 Function vs Equation 3 Value of Polynomial 4 Algebraic Operation 4.1 Division of Polynomial 4.1.1 Example 4.1.2 Degree 5 Exercise 1 6 Equality of Polynomials 6.1 Substituting Suitable Values 6.1.1 Example 6.2 Comparing Coefficients 6.2.1 Identifying contributing terms 6.2.2 Example 6.3 Combination of both Methods 7 Exercise 2 8 Divisible Polynomial 8.1 Factorizing Cubic Polynomial 9 Exercise 3

Notes

On the surface, factor theorem & remainder theorem seems easy when compared to other subtopics. However, a lot of student make the mistake and think that this subtopic is about CALCULATION. It is NOT (calculator will handle most of the calculations anyway). It is about THEOREM and how to USE it CORRECTLY. A correct calculation with the wrong way of using the theorem will result in LESS/ZERO marks. There will also be a lot of WRITING (of words, not just numbers and symbols), which is another thing student find it hard to do. Failure to learn the usage of the relatively simple theorems here will also probably mean failure to grasp many other subtopics in the syllabus.

Learning Objectives (Syllabus)

• understand the meaning of the degrees and coefficients of polynomials
• carry out elementary operations on polynomials
• use the condition for the equality of two polynomials
• find the factors and zeroes of polynomials
• prove and use the remainder and factor theorems

Prior Knowledge

• Basic algebraic skills

Degree & Coefficients

where is a positive integer.

• degree  : refers to the highest power/index of the polynomial
• terms  :
• coefficients  :

Example

• is a polynomial of degree
• How do we refer to the and ?
  • Coefficient of (of the term in
  • Term in
• Simple enough, but MAKE SURE you don't give term when asked for coefficient or vice versa.
• • Coefficient of
  • Term in
  • Coefficient of Huh? Where got ?
  • How about the ? We just refer to it as the constant term

Common Names

• linear - degree 1 Ex:
• quadratic - degree 2 Ex:
• cubic - degree 3 Ex:

Function vs Equation

Polynomials is a type of function and NOT equation.

Example

• is a function. We can find which are the values of the function

• is a equation, and we can actually find its solution, which is

Value of Polynomial

The value of a polynomial when

• is written as
• and is calculated by substituting into the polynomial
• NOTE: Leave the arithmetic to the calculators, especially when it involves fraction. And calculate it by entering the formula and then only enter the value of

Example:

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Algebraic Operation

We can apply addition, subtraction and multiplication to polynomials by using basic algebra laws.

Example:

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Note : If we multiply polynomial of degree with polynomial of degree , we get polynomial of degree

Division of Polynomial

This is where the fun begins. :P Before we look at division of polynomials, lets revise division of numbers. Yes, the method (known as long division) that we learn in primary school

• 
• Which number write in and which one out?
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• We will next take the and MINUS the (we will normally NOT write the minus sign, but lets do it here)
• 
• Thus,
• We can also rearrange it to
• Once again, from or
• Let's revise the terms used to describe these numbers
  • The is the divisor
  • The is the quotient
  • The is the remainder

Example

• 
• The one thing to keep in mind is that the steps are the same as when it involves numbers
  • So the the first thing to do is to write the polynomial and the divisor at its right place
  • 
  • The next step will be looking at
  • So
  • However, keep in mind we must multiply with , not just
  • 
  • Just like in numbers, we will then MINUS that product. My advice is, do actually write the minus sign, AND wrap that product in BRACKETS.
  • 
  • Let's minus it
  • 
  • Bring down the next term
  • Next, look at Thus
  • 
  • 
  • • Divisor :
    • Quotient :
    • Remainder :
    • 
    • 
    • You will notice that expanding the LHS will simply give us back the RHS, so this can be a method to check.
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  • Oops... How do we subtract from ? Let's look at example of numbers, like , whereby and so on. We can see the 's playing an important part as place holders.
  • For the polynomial, we can either leave some space in between, or to actually fill in the zero coefficients. Let's restart.
  • or just something like
  • 
  • The minus sign and brackets are very important here to avoid careless mistakes
  • You can omit the if you want to here
  • 
  • 
  • 
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  • • Divisor :
    • Quotient :
    • Remainder :
    • 
    • 

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• A somewhat simpler type but easy to make careless mistakes when writing the result
  • 
  • 
  • 

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• • Eh? Wait a minute....
  • If you spot the mistake, good for you. If you didn't, just keep in mind this is a VERY common careless mistake when we do things too quickly (it has something to do with our brain always looking for the simplest situation/solution)
  • But ?
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  • 

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  • Hmmm... no good. Better.
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  • It seems we can't further. That means in this case, the remainder is not just a number, but a linear polynomial/function
  • Try rewriting the original polynomial directly (this would be the most common use of long division in this subtopic)
  • 

• 
• It's impossible to do it by (Unless you want to divide it twice!)
  • So first we expand the divisor
  • You should probably try this on your own
  • Answer :
  • 

Degree

• When we divide polynomial of degree with polynomial of degree , we get
  • Let's look at the examples we have done to give us an idea
    • 
    • 
    • 
  • Quotient of degree
  • Remainder or less

Exercise 1

Using long division, find the quotient & remainder when the polynomial is divided by the given divisor, and rewrite the polynomial

1)

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2)

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3)

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4)

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5)

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6)

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7)

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Equality of Polynomials

Two polynomials are equal only if all coefficients are the same

For example, if

• Note:
  • Equal polynomials are identity (thus the sign, but using is still acceptable in working), it holds for all values of
  • we DO NOT, however solve for the value of

Substituting Suitable Values

If we are given two polynomials (albeit written in different form), we can substitute ANY value of to help us find any unknowns. Thus

• Use values which will make some of the terms to be zero
• Use other (easy to count) values if needed ( )
• Quickly check by substituting other suitable values

Example

• 
• The thing to keep in mind is we can substitute ANY value of , we choose the easiest to calculate
  • Since have , we substitute values that will make the terms zero
  • To make zero, we use
  • To make zero, we use
  • We then use some other simple values to check, in this case easiest to use
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  • We need to substitute BOTH sides, and also the term that becomes zero need not be written, of course
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• We check by substituting
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  • (Use calculator for quick calculation)

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• I can't stress how important it is to check, especially here where
  • we deal with fractions and negatives
  • subsequent parts of the question will likely require the values found
  • the method of checking is very simple and doesn't take ten seconds

• 
• Note that
  • Thus, when we put , both and will be zero
• Try this on your own first! Answer
• 
• Remember to check!

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• What else can we put to make it any term zero?
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• What else? No other values can make the terms zero, but since we already have two unknowns, we can use any other easy-to-count values.
• (you can also use , I just don't like dealing with negative)
  • 
  • 
  • 
• Check with (since we can't use or )
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  • 

Comparing Coefficients

• When no suitable values can be substituted, we use comparison of coefficients
• Always try to compare terms which has less contributing terms (highest power & constant)

Identifying contributing terms

• 
  • term in will come from
  • term in will come from
  • constant from

• 
  • term in will come from
  • term in will come from
  • term in will come from
  • constant from
• Thus, it's normally easier to look at the highest term and the constant first.

Example

• 
• First, note that there are no simple value of that we can substitute to make any term to be zero, so we will do comparison of coefficients
  • We can compare the coefficients of and constant. However we will only need two since there are only two unknowns. Thus, we will choose to compare coefficient of and constant
  • Let's look at what will happen for coefficient of
    • On LHS we have
    • On RHS , which means the term in is
  • Comparing coefficient of  :
    • Note : We DON'T write , we are only comparing the coefficients, so we can only write the coefficients, not the terms
  • For constant
    • On LHS :
    • On RHS :
  • Comparing constant:

• 
• If its too difficult to see the contributing terms, we can expand the . However, it is counterproductive to expand and rearrange the whole of RHS
  • 
  • Comparing coefficients of  :
  • Comparing constant:
    • We can't solve this yet. So let's compare coefficient of also
  • Comparing coefficients of  :
    • 
  • 

Combination of both Methods

• Always do substitution BEFORE comparing since substitution is easier

Example

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• We can substitute, so do it first
  • 
  • No other easy-to-count values to substitute, so compare coefficients
  • Compare coefficient of term in :
  • Compare constant :
  • Finished? No, check.
    • 
    • 

Exercise 2

Find the values of

1)

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  • Check :

2)

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  • Check :

3)

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  • Check :

4)

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  • Check :

5)

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  • Check :

6)

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  • Check :

7)

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  • Check :

8)

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  • Check :

9)

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  • Check :

10)

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  • Check :

Divisible Polynomial

• When there is no remainder (remainder is zero), we say that the polynomial can be divided or is divisible by the divisor.
  • Which also means we can just write

• If polynomial of degree is divisible with polynomial of degree , we get a polynomial of degree
• For example , degree of is

Factorizing Cubic Polynomial

Lets say we know that can be divided by , and we need to find the quotient. The logical thing to do is to do long division (there will be no remainder), but there is a faster method.

• First, we note that the quotient will be of degree
• Thus, we know that
• However, if we write , we end up having to write the solution for three unknowns. So instead, we try to work it out in our heads directly.
  • We know its going to be a quadratic, so let's look at it term by term
  • 
  • The on the left hand side comes from Meaning is or just simply
  • Similarly, for the constant, Thus is
  • So now we have
  • Meaning
  • To find the coefficient of we are left with two choice, the or the on the LHS. Which one to choose? And do we need both? Use to find, use to check (or vice versa)
  • Where will the come from? , but also
  • In other words,
  • Thus is
  • 
  • Check the Correct.
  • Alternatively, we can opt to do calculation by hand.
    • 
    • Comparing coefficient of  :
    • 
    • Check :  :

Note :

• This is a faster method, however, there is nothing wrong going back to long division if somehow you just can't work it out (especially during exams)
• You MUST, however, check. (since you won't get be able to see the zero remainder like in long division which will help us confirm it is correct)
• Only employ this method for cubic polynomials. Don't do this for higher degrees (quartic and so on), since you will likely end up spending more time (and more likely to make mistakes) compared to long division.

Exercise 3

Complete the following

1)

2)

3)

4)