Leaving Certificate Mathematics/Algebra

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.

Algebra 1Edit

  1. Expressions - The very basics; Addition, Subtraction, Multiplication, Division, Algebraic Notation, and using Pascal's Triangle.
  2. Factorising - Finding the factors of an expression by using the Highest Common Factor (HCF), Grouping, Difference of Two Squares, Difference of Two Cubes, and Quadratic Trinomials.
  3. Algebraic Fractions - Addition and Subtraction, and Multiplication and Division of algebraic fractions.
  4. Binomial Expansions - A simpler way of expanding an expression of two terms, when they are to a high power.
  5. Binomial Terms - Finding a term at a specific location, or finding the location at which a variable is to a certain power.

Exam QuestionsEdit


Paper 1 Question 1Edit

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 Question2Edit

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 1Edit

(a) Express   in the form   where   and  .


(i) Let   where   is a constant Given that   is a factor of   find the value of  

(ii) Show that   simplifies to a constant.


(i) Show that  .

(ii) Hence, or otherwise, find, in terms of   and  , the three values of   for which  .

Paper 1 Question 2Edit

(a) Solve without using a calculator, the following simultaneous equations:






Solve the inequality   where   and  


the roots of   are   and   where  .

Find the quadratic equation whose roots are   and  .




Show that there exists a real number   such that for all  



Show that for any real values of   the quadratic equation


has real roots.


Paper 1 Question 1Edit

(a) Solve the simultaneous equations:




(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 2Edit

(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 1Edit

(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 2Edit

(a) Solve the simultaneous equations




(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  .