Algebra

What is algebra? Why are there letters in the equation? This page links to resources that will help you answer these questions.

Algebraic expressions involve pronumerals (letters) to represent values. Pronumerals can take many different values. We often need to plug our own values into a given formula for calculating our finances, and this is called substitution.
There are rules around adding, subtracting, multiplying and dividing with pronumerals. Review these pages and videos to see how it works.

• A1.1 Algebraic operations
  Introducing the basic skills for addition, subtraction, multiplication and division (+ - × ÷) of algebraic expressions. Courses in science, engineering and other fields require you to have these skills.
• A1.2 Algebraic substitution
  Find out how to replace pronumerals with numbers in a formula to get a numerical value for some quantity. In many courses you will be required to use formulae to calculate something of interest. The process of substituting numbers for pro-numerals in an expression or formula is called substitution.
• A1.3 Removing brackets
  What do I do with the brackets? Brackets are useful to group numbers, pronumerals and operations together as a whole. Whatever is around the brackets affects all the things inside the brackets.
• A1.4 Algebraic fractions – Addition and subtraction
  How do I deal with fractions involving pronumerals? Adding and subtracting fractions always requires a common denominator (which is the lower half of the fraction). These need to be the same before you can add or subtract. This means converting some of the fractions to forms that are consistent in the denominators. It is possible to do this even when the fractions include pronumerals.
• A1.5 Algebraic fractions – Multiplication and division
  What is an algebraic fraction? The numerator (top) or denominator (bottom) of a fraction can be in algebraic form involving numbers and variables (represented by pronumerals or letters). We cover the multiplication and division of fractions containing algebraic terms.
• A1.6 Quadratics
  Let's take a look at algebraic fractions where the denominator is a quadratic expression. Such fractions are common in mathematics and engineering.
• A1.7 Partial fractions
  Find out how to express an algebraic fraction as a sum of simpler algebraic fractions (partial fractions). The method of partial fractions involves breaking up an algebraic fraction into simpler parts that are added together. This is useful in integration and in finding inverse Laplace and Fourier transforms.
• A2.1 Rearranging formulae
  Rearranging formulas, also called transposition of formulas, is a necessary skill for most courses. Let's work on some essential skills in manipulating formulas.
• A2.2 Rearranging formulas – Brackets and fractions
  Learn how to manipulate or rearrange formulas that involve fractions and brackets.
• A2.3 Transposition of formulas with challenges
  Are you still trying to get that variable on its own from the formula, but it is in a tricky place – or maybe it appears more than once? Here we demonstrate manipulating or rearranging complex formulas, with overviews and practice questions.
• A3.1 Common factors
  A common factor is a number or pronumeral that is common to terms in an algebraic expression. Removing the common factors allows us to factorise algebraic expressions and write them in a simpler form. Factorisation using common factors is a basic skill in mathematics so we'll discuss that too.
• A3.2 Perfect squares
  A perfect square is something like \(5^{2}, x^{2}\) or \(\left(x+1\right)^{2}\). In this module we deal with perfect squares of the form: \(\left(x\pm y\right)^{2} =x^{2}\pm2xy+y^{2}.\)These forms are very common in mathematics and help us simplify expressions. They are important to know if you are an engineering or science student.
• A3.3 Difference of two squares
  Introducing the difference of two squares formula which allows us to factorise expressions like \(a^2 - b^2\) or \(x^2-36\).
• A3.4 Quadratics
  Quadratic expressions have the general form \[ax^2+bx+c\] where \(a\), \(b\) and \(c\) are real numbers and \(a\neq 0\). Quadratics frequently arise in mathematics, science and engineering. Find out how to factorise a quadratic into two linear factors. For example \[x^2+5x+6 = \left(x+2\right)\left(x+3\right).\]
• A3.5 Completing the square
  Quadratic expressions have the general form \[ax^2+bx+c\] where \(a\), \(b\) and \(c\) are real numbers and \(a\neq 0\). Quadratics frequently arise in mathematics, science and engineering. We explain how to factorise a quadratic into two linear factors using the completing the square method. This is a general method that allows any quadratic to be factorised. For example \[x^2+8x-5 = \left(x+4-\sqrt{21}\right)\left(x+4+\sqrt{21}\right).\]
• A3.6 Polynomial long division
  You may have learnt long division in school. It enables you to divide two numbers that may be quite large. It is also possible to use long division on polynomials so that they can be factorised. Cubic polynomials are the most common. Find out how to write \[x^3-5x^2-2x+24=\left(x-4\right)\left(x-3\right)\left(x+2\right).\] This type of factorisation is useful for graphing the cubic polynomial.