1) The document discusses polynomials, including adding, subtracting, multiplying, and dividing polynomials. It also covers finding the roots and zeros of polynomials. 2) Partial fraction decomposition is explained for denominators that are linear factors, repeated linear factors, quadratic factors, and repeated quadratic factors. 3) Key concepts covered include the remainder and factor theorems, zeros of polynomials, and converting improper rational expressions to proper rational expressions using long division and partial fraction decomposition.

1. CHAPTER 5:
POLYNOMIALS

2. • Perform addition, subtraction, multiplication and division of
polynomials.
• Use the remainder and factor theorems in problem solving.
• Find the roots and zeros of a polynomial.
• Perform partial fraction decomposition when the denominators are
in the form of:
– A linear factor.
– A repeated linear factor.
– A quadratic factor
At the end of this chapter, student should be able to:

3. Polynomials
1 2
1 2 1 0
0
Polynomial function ( ) is
( ) .....
where,
- leading coefficient, 0
- degree of the polynomial & positive integer
- constant term
n n
n n
n n
P x
P x a x a x a x a x a
a a
n
a


     


5. prepared by NAMN

6. prepared by NAMN

8. Steps to be taken:
S1 Divide the 1st term of numerator, P(x) by the 1st term of denominator,
D(x) answer, Q(x).
S2.Multiply the denominator, D(x) by the answer and put below
numerator.
S3 Subtract to create a new polynomial.
S4.Repeat S1 using the new polynomial until the degree of new
polynomial is less than denominator.

9. Example
2
3 2
Multiply:(2 3)(3 2 3)
4 5 2
Use long division to find
2
x x x
x x x
x
  
  


10. Remainder Theorem
Example:
refer example 1 and 2 in textbook page 234.
Note: If a polynomial is divided by a quadratic expression, then
the remainder, R = Ax + B. Where A and B are constant to be
determined.
If R is the remainder after dividing
the polynomial P(x) at (x-a), then
P(a)=R

11. Factor Theorem
For a polynomial P(x) and a constant a,
iff P(a) = 0, then (x - a) is a factor of
P(x).

12. Zeros of Polynomials
The zeros of the polynomial can be obtained
when P(x) is completely factorised and then
solved for zero.
Therefore a, b and c are zeros of the
polynomial
P(x).
If ( ) ( )( )( ),
then ( ) 0, ( ) 0 and ( ) 0.
P x x a x b x c
P a P b P c
   
  

13. 3 2
3 2
If ( ) ( )( )( ),
then , and are called the roots of
the polynomial equation ( ) 0
Example:
( ) 2 5 6
( ) ( 1)( 2)( 3)
then the zeros are 1,-2 and 3.
1 is a root of ( ) 2 5
P x x a x b x c
x a b c
P x
P x x x x
P x x x x
x P x x x
   


   
   
    6
since (1) 0.
x
P



14. Partial fraction
If the degree of P(x) is less than that of D(x), then
is called a proper fraction.
Only a proper rational expression can be expressed as
partial fractions.
( )
( )
P x
D x
partial fraction
decomposition
Non-
repeated
Linear
Repeated
linear
Non-
repeated
quadratic
Repeated
quadratic

15. Partial fractions decomposition
Case 1:Denominator consists of non-repeated linear factors.
 Contain an expression of the form for each
non-repeated linear factor (ax+b) in the denominator.
A
ax b


16. Partial fractions decomposition
Case 2: Denominator consists repeated linear
factors.
 Contain an expression of the form
for each repeated linear factor of multiplicity n.
   
2
1
2
..... n
n
A A
A
ax b ax b ax b
  
  

17. Partial fractions decomposition
Case 3: Denominator consists of non-repeated quadratic
factors.
 If a non-reducible factor, occur
in the denominator, then the partial fraction
corresponding to this factor is
2
ax bx c
 
2
Ax B
ax bx c

 

18. Partial fractions decomposition
Case 4: Denominator consists of repeated
quadratic factors.
 If the factor is repeated twice
in the denominator, then the form of the
partial fractions corresponding to this would
be
2
ax bx c
 
 
2
2 2
Ax B Cx D
ax bx c ax bx c
 

   

19. Improper rational expression
Improper rational expression is when the
degree
of P(x) greater than D(x).
S1: long division
S2: partial fraction
reduce the
improper rational
expression to proper
rational expression