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RMIT University Library - Learning Lab



What is algebra? Why are there letters in the equation?
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

    This section introduces the basic skills for addition, subtraction, multiplication and division (+ - × ÷) of algebraic expressions.

  • A1.2 Algebraic substitution

    This module shows how to replace pronumerals with numbers in a formula to get a numerical value for some quantity.

  • 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. This module explains how to work with 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. This module discusses how this is done.

  • 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). In this module we cover the multiplication and division of fractions containing algebraic terms.

  • A1.6 Quadratics

    This module looks at algebraic fractions where the denominator is a quadratic expression. Such fractions are common in mathematics and engineering.

  • A1.7 Partial fractions

    This module explains how to express an algebraic fraction as a sum of simpler algebraic fractions (partial fractions). This is of use 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. This module provides 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. This includes 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 and is discussed in this module.

  • 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

    This module introduces 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. This module explains 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. This module explains 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. This module explains how to do this. Cubic polynomials are the most common. You will see 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.