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Elementary Algebra (i)
(1) Topics coverec in this sheet
(2) Recall
(3) Exercises
Note: The solutions for the qestion no. 3b may be dowloaded from
here.
(1) Topics covered in this sheet:

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/nonreal roots
(2) Recall:
(a) Nomenclature
(i) Symbols
(ii) Sets
(iii) Numbers
(iv) Note that:
(b) Sums/Differences of two squares/cubes.
(c) Definition:
We say that f(x) is a polynomial in x
of order n when
and .
(d) Factor Theorem:

If (xa) 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 (xa) is a factor of f(x).
(e) Real/nonreal factors/roots
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.
(3) Qustions:
(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) 

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