Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the Persian mathematician, astronomer, astrologer and geographer, Muhammad bin Mūsā al-Khwārizmī titled Kitab al-Jabr wa-l-Muqabala meaning "The Compendious Book on Calculation by Completion and Balancing", which provided symbolic operations for the systematic solution of linear and quadratic equations. Al-Khwarizimi's book made its way to Europe and was translated into Latin as Liber algebrae et almucabala.
Together with geometry, analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics. Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variables, definition of polynomials, along with factorization and determining their roots.
Algebra is much broader than elementary algebra and can be generalized. In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.
Exam Questions[edit]
Paper 1 Question 1[edit]
1. (a) Express the following as a single fraction in its simplest form:

(b) (i)
where 
Given that
is a real number such that
, prove that
is a factor of
.
(ii) Show that
is a factor of
and find the other factor.
(c) The real roots of
differ by
where
and
.
(i) Show that
.
(ii) Given that one root is greater than zero and the other root is less than zero, find the range of possible values of
.
Paper 1 Question2[edit]
2. (a) Solve the simultaneous equations:


(b) (i) Solve for x:

(ii) Given that
is a factor of
where 
find the value of
and the value of
.
(c) (i) Solve for y: 
(ii) Given that {math>\ x = \alpha</math> and
are the solutions of the quadratic equation
where
and 
show that
is independent of
and
.
Paper 1 Question 1[edit]
(a) Express
in the form
where
and
.
(b)
(i) Let
where
is a constant Given that
is a factor of
find the value of 
(ii) Show that
simplifies to a constant.
(c)
(i) Show that
.
(ii) Hence, or otherwise, find, in terms of
and
, the three values of
for which
.
Paper 1 Question 2[edit]
(a) Solve without using a calculator, the following simultaneous equations:



(b)
(i)
Solve the inequality
where
and 
(ii)
the roots of
are
and
where
.
Find the quadratic equation whose roots are
and
.
(c)
(i)
for 
Show that there exists a real number
such that for all 

(ii)
Show that for any real values of
the quadratic equation

has real roots.
Paper 1 Question 1[edit]
(a) Solve the simultaneous equations:


(b)
(i) Exspress
in the form
where 
(ii) Let
.
Show that
is a factor of
.
(c)
is a factor of 
Show that 
Exspress the roots of
in terms of p
Paper 1 Question 2[edit]
(a) Solve for x
where 
(b) The cubic equation
has one integer root and two irrational roots. Exspess the rational roots in simplest surd form.
(c) Let
wher
and
are constants and 
(i) show that
.
(ii)
and
are real numbers such that
and
. Show that if
, then 
Paper 1 Question 1[edit]
(a) Find the real number a such that for all
,

(b)
, where
and
are constants. Given that
and
are factors of
, find the value of
and the value of
.
(c)
is a factor of
.
(i) Show that
.
(ii) Express the roots of
in terms of
and
.
Paper 1 Question 2[edit]
(a) Solve the simultaneous equations


(b)
(i) Find the range of values of
for which the quadratic equation

(ii) Explain why the roots are real when t is an integer.
(c)
and
, where
is a positive real number. Find, in terms of
, the value of
for which
.