IAS Maths Notes
Pure 2
, 11
Algebraic Fractions
Algebraic fractions can be simplified using division
-
Example
①
simplify
a)7x
"
- 2x + Ex
b) xi + 7x + 12
* (x + 3)
- -+ =
(x + 3)(x + 4)
(x + 3)
=
7x3 -
2x7 + 6 xx+ 4
=
1.2 Dividing Polynomials
A
polynomial is
expression With
positive whole number indices
·
an
division used divide
Long can be to
polynomials by :
-
> (X + b) ,
where b is a constant
< (ax + b) ,
where a sb are Constants
Example
① Divide 23 2x2-17x (x 3)
+
by -
x2 + 5x -
2
X -
3 23 + 2x2 -
17x + 6
- x3 -
342
-
5x2 1x -
-
5x2- 15x v
-
25 + 6
2x
+
--
1 3 The
.
Factor Theorem
If f(x) is
polynomial then :
·
a
>
If F(p) = 0 , then (xp) is a
factor of F(x)
>
If (x-p) is a factor of f(x), then
F(p) 0
=
>
If f(E) =
0 , then (ax-b) is a factor of fix)
>
If (ax- b) is a
factor of f(x) then (a) = 0
Example
① Show X-2 is a
factor Xx + x2-4x-4
of
f(z) = 23 + 23 -
4(2) 4 -
= O : CX-2) is a
factor
, 1 4 The
.
remainder theorem
f(x) divided by Cax-b) then
If a
polynomial is the remainder
·
is F(
Example
① Find the remainder when -20x + 3 is divided by X-4
f(z) = f(Y) =
f(y)
f(y) = 43 20(4) -
+ 3
f(4) = -
13
remainder is-13
1 5 .
Mathematical Proof
·
A
proof is a logical and structured argument to show that a
mathematical statement or
conjecture is always true
Theorem is mathematical fact statement
proven
·
a or
·
Conjecture is a mathematical theory yet to be
prover
Mathematical
proof usually starts with previously established
-
theorems and then logical steps
works
through a series of
The final step in a
proof
is a statement
-
known facts statement
/theorems
>
clearly shown >
logical steps of Proof
Mathematical Statements be dealuction
can
proven using
·
-
Procedure for deduction
1) Identify known facts / theorems
11)Conduct
logical step-wise calculations
111) State the statement
proven
Example
D Prove that of two odd odd
the
product numbers is
=
(2x 1)(2y + 1)
+
=
(xy + 2x + 2y) + 1 factor out z
=
z(zxy + x +
y) + 1 z(n) + 1 makes an odd number
The odd odd
product two is
: numbers
of any
Pure 2
, 11
Algebraic Fractions
Algebraic fractions can be simplified using division
-
Example
①
simplify
a)7x
"
- 2x + Ex
b) xi + 7x + 12
* (x + 3)
- -+ =
(x + 3)(x + 4)
(x + 3)
=
7x3 -
2x7 + 6 xx+ 4
=
1.2 Dividing Polynomials
A
polynomial is
expression With
positive whole number indices
·
an
division used divide
Long can be to
polynomials by :
-
> (X + b) ,
where b is a constant
< (ax + b) ,
where a sb are Constants
Example
① Divide 23 2x2-17x (x 3)
+
by -
x2 + 5x -
2
X -
3 23 + 2x2 -
17x + 6
- x3 -
342
-
5x2 1x -
-
5x2- 15x v
-
25 + 6
2x
+
--
1 3 The
.
Factor Theorem
If f(x) is
polynomial then :
·
a
>
If F(p) = 0 , then (xp) is a
factor of F(x)
>
If (x-p) is a factor of f(x), then
F(p) 0
=
>
If f(E) =
0 , then (ax-b) is a factor of fix)
>
If (ax- b) is a
factor of f(x) then (a) = 0
Example
① Show X-2 is a
factor Xx + x2-4x-4
of
f(z) = 23 + 23 -
4(2) 4 -
= O : CX-2) is a
factor
, 1 4 The
.
remainder theorem
f(x) divided by Cax-b) then
If a
polynomial is the remainder
·
is F(
Example
① Find the remainder when -20x + 3 is divided by X-4
f(z) = f(Y) =
f(y)
f(y) = 43 20(4) -
+ 3
f(4) = -
13
remainder is-13
1 5 .
Mathematical Proof
·
A
proof is a logical and structured argument to show that a
mathematical statement or
conjecture is always true
Theorem is mathematical fact statement
proven
·
a or
·
Conjecture is a mathematical theory yet to be
prover
Mathematical
proof usually starts with previously established
-
theorems and then logical steps
works
through a series of
The final step in a
proof
is a statement
-
known facts statement
/theorems
>
clearly shown >
logical steps of Proof
Mathematical Statements be dealuction
can
proven using
·
-
Procedure for deduction
1) Identify known facts / theorems
11)Conduct
logical step-wise calculations
111) State the statement
proven
Example
D Prove that of two odd odd
the
product numbers is
=
(2x 1)(2y + 1)
+
=
(xy + 2x + 2y) + 1 factor out z
=
z(zxy + x +
y) + 1 z(n) + 1 makes an odd number
The odd odd
product two is
: numbers
of any
