Dr. Joseph N. Grima, Department of Chemistry
University of Malta, Msida, MSD 06, MALTA
http://staff.um.edu.mt/jgri1/teaching/ch033

• Introduction to the f(x) notation
• Factorisation
• Quadratics
• Difference of two squares
• Difference and sum of of two cubes
• Factor theorem
• Soving simple equations
• The concept of real/non-real roots

(a) Nomenclature

 
(i) Symbols

(ii) Sets

(iii) Numbers
 

(iv) Note that:

We say that  f(x) is a polynomial in x of order n when 

and .

• If (x-a) is a factor of f(x) then f(a) = 0.
• Conversly, for a given f(x) = a polynomial in x, if f(a) = 0 then (x-a) is a factor of f(x).

Consider f(x) a  n^th order polynomial in x with rael cofficients c_i given by:

• f(x) will have a maxium of n real roots and n real factors.
• f(x) may also have some complex roots. If this is the case, these complex roots must occur in pairs of complex conjugates. The total number of real/complex roots is always equal to n (taking into considerations any double roots!). Thus for example: 
• A cubic may have: (i) 3 real roots, or (ii) 1 real root and two complex roots which have to be complex conjgates. 
• A quardic may have: (i) four real roots, (ii)  2 real roots and 2 complex roots which have to be complex conjgates, or (iii) 4 complex roots, i.e. two pairs of complex conjgates.
• When n is ODD, we are guanateed atb least one real root. This is not the case when n is EVEN.

(a) Evaluate (i)  f (2) , (ii)  f (-1), (iii) f (-3)  and (iv)  f (0);
(b) Factorise f(x) as completly as possible.
(c) Solve the equations f(x) = 0
... given that f(x) is given by:
 

 

	No.	f (x)
	1
	2
	3
	4
	5
	6
	7
	8
	9
	10
	11
	12
	13
	14
	15
	16
	17
	18
	19
	20
	21
	22
	23
	24
	25
	26
	27
	28
	29
	30
	31
	32
	33
	34
	35
	36
	37
	38
	39
	40
	41
	42
	43
	44
	45
	46
	47
	48
	49
	50
	51
	52
	53
	54
	55
	56
	57
	58
	59
	60