1. Chapter 5
Additional Maths
Adam Gibson

2. Identities and Equations
Look at this equation:
Always true
An equation tells you that something is true in a particular situation, not always!
Now look at this equation:
Always true
Certainly! The LHS and RHS are identically equal
This is called an “identity”

3. Polynomials is not a polynomial
… are mathematical expressions of the form:
The above expression represents a polynomial in x of nth order, or an “nth degree polynomial”.
What is a zero degree polynomial?
A constant
What kind of polynomial is a quadratic?
A second degree polynomial
Is this a polynomial in x? What order is it?
For the 4th order example given, ask students to give values of the ak coefficients
The graph is a good subject for discussion of the nature of higher order polynomials
Don’t forget to ask the students what a 1st order polynomial is.
Yes, degree 4
is not a polynomial

4. Some basics descending order ascending order unordered
A polynomial of zero degree is a constant
A polynomial of degree 1 is called linear
e.g. -
A polynomial of degree 2 is called quadratic
e.g. –
Degree 3 – cubic, 4 – quartic, 5 – quintic etc.
“Degree” is also sometimes called “order”
descending order
ascending order
unordered

5. Equating coefficients
If: -
This may be obvious, but it’s very useful!
Example:
Identical polynomials have identical coefficients

6. ÷ The algebra of polynomials
Polynomials can be added, subtracted or multiplied quite easily.
Division is harder!
Problem:
Divide By ÷
Review examples of multiplication in particular, and show that it may be necessary to include an intermediate step because it is quite complicated.
Students need to think about the degree of polynomial products, hopefully it will be clear with a couple of examples.
The correct method for finding the solution requires thinking about two things: the degree of a product of two polynomials, and equating coefficients

7. 2 1 To find the solution, use this identity: has degree 4
(see textbook p.9)
has degree 4
is the divisor and has degree 2
The quotient, q(x), must have degree:
The remainder, r(x), must have degree: 2 1
The identity can therefore be written in the following form:

8. What next? Equate coefficients!
3 = B
0 = C - 8A
-5 = G - 8B
5 = H – 8C
C=16, G=19, H=133
so, finally we get:

9. The “remainder” and “factor” theorems
Let’s return to the formula:
Now let’s consider a linear divisor:
What can we say about r(x)?
A: It will be a constant, so:

10. The remainder theorem contd.
Setting x=alpha:
What does this prove?
It proves that if
is a factor of a(x), then a(α) is zero.
You can say
“divides a(x) exactly”.
If (x-3) divides p(x) exactly, what does it
tell you about the graph of p(x)?
It tells you that 3 is a root of the equation p(x)=0,
or equivalently that the graph cuts the x-axis at 3.

11. This tomb holds Diophantus. Ah, what a marvel!
And the tomb tells scientifically the measure of his life.
God vouchsafed that he should be a boy for the sixth part
of his life; when a twelfth was added, his cheeks acquired
a beard; He kindled for him the light of marriage after
another seventh, and in the fifth year after his marriage
He granted him a son.
Alas! late-begotten and miserable child, when he had
reached the measure of half his father's life, the chill
grave took him. After consoling his grief by this
science of numbers for four years,
he reached the end of his life.
How old was Diophantus when he died?

13. Diophantine equations
Diophantus only looked
for positive integer
solutions; he considered
negative and irrational
solutions as foolish.
the smallest solutions are:
Diophantus of Alexandria - Διόφαντος ο Αλεξανδρεύς –
was a Greek mathematician. He was known for his study of
equations with variables which take on integer values
and these Diophantine equations are named after him.
Diophantus is sometimes known as the "father of Algebra".
He wrote a total of thirteen books on these equations.
Diophantus also wrote a treatise on polygonal numbers.

14. Diophantine equations
Examples:
Bezout’s identity
It can be shown that if a and b are integers, and d is the
greatest common divisor of a and b, then there exist
solutions for x and y to this equation.
This is a linear Diophantine equation.
n=2; Pythagorean triples. n>2,
no solutions.
The study of Diophantine equations is part of number
theory and is actually a very advanced topic.