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CIE Centre A-level Pure
Maths
P2 Chapter 1
© Adam Gibson
Identities and Equations
Look at this equation:
x  3 x
Always true
An equation tells you that something is true in a particular
situation, not always!
Now look at this equation:
x2  y 2  ( x  y)( x  y)
Always true
Certainly! The LHS and RHS are identically equal
x  y  ( x  y)( x  y)
2
2
This is called
an “identity”
Polynomials
… are mathematical expressions of the form:

n
a
x
k
k 0
The above expression represents a polynomial in x of nth
degree, or an “nth degree polynomial”.
What is a 0-degree polynomial?
A constant
What kind of polynomial is a quadratic?
A second degree polynomial
Is this a polynomial in x? What degree is it?
f ( x)  x 2  16 x  5  x 4
Yes, 4th degree
y  sin x  x3 is not a polynomial
k
Some basics
•A polynomial of degree zero is a constant
•A polynomial of degree 1 is called linear
e.g. - f ( x )  3 x  8
•A polynomial of degree 2 is called quadratic
2
e.g. – f ( x)  7 x  70 x  5
“Degree” is also
sometimes called
“order”
•Degree 3 – cubic, 4 – quartic, 5 – quintic etc.
f ( x)  8x6  9 x5  x4  11x3  2 x 2  x  30 descending order
f ( x)  30  x  2 x 2  11x3  x 4  9 x5  8x6 ascending order
f ( x)  2 x2  8x6  11x3  x 4  30  x  9 x5
unordered
Equating coefficients
n
n
k 0
k 0
k
k
F
(
x
)

a
x

G
(
x
)

b
x
If:  k
k
 k , ak  bk
This may be obvious, but it’s very useful!
Example:
3x3  4 x 2  10 x  60  Ax3  Bx 2  Cx  D
 A  3, B  4, C  10, D  60
Identical polynomials have identical coefficients
The algebra of polynomials
•Polynomials can be added, subtracted or multiplied
quite easily.
•Division is harder!
÷
Problem:
Divide
By
2 x 4  3x3  5 x  5
x2  8
The correct method for finding the solution requires thinking about two
things: the degree of a product of two polynomials, and equating
coefficients
To find the solution, use this identity:
a ( x )  b( x ) q ( x )  r ( x )
a ( x)  2 x  3x  5 x  5
4
b( x )  x 2  8
3
(see textbook p.9)
has degree 4
is the divisor and has degree 2
•The quotient, q(x), must have degree:
2
•The remainder, r(x), must have degree:
1
The identity can therefore be
written in the following form:
2 x4  3x3  5x  5  ( x 2  8)( Ax 2  Bx  C )  Gx  H
What next? Equate coefficients!
2=A
3=B
q( x)  2 x  3x  16, r ( x)  19 x  133
2
0 = C - 8A
-5 = G - 8B
5 = H – 8C
so, finally we get:
a ( x)  2 x 4  3 x 3  5 x  5
 ( x  8)(2 x  3x  16)  19 x  133
2
2
The “remainder” and “factor” theorems
Let’s return to the formula:
a ( x )  b( x ) q ( x )  r ( x )
Now let’s consider a linear divisor:
a ( x)  ( x   )q ( x)  r ( x)
What can we say about r(x)?
A: It will be a constant (degree zero), so:
a ( x)  ( x   )q ( x)  R
The remainder theorem contd.
Setting x=alpha:
What does this prove?
a ( )  0  R
It proves that if
x 
is a factor of a(x), then a(α) is zero.
You can say
x 
“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.
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?
Diophantine equations
x  94 y  1 , x, y 
2
2
the smallest solutions are:
x  2,143, 295, y  221, 064
Diophantus only looked
for positive integer
solutions; he considered
negative and irrational
solutions as foolish.
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.
Diophantine equations
Examples:
ax  by  d
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.
xn  y n  z n
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.