Menu

Factorising - Revision Notes

Please provide a rating, it takes seconds and helps us to keep this resource free for all to use

[ 6 Votes ]

In addition to the revision notes for Factorising on this page, you can also access the following Expressions learning resources for Factorising

Expressions Learning Material
Tutorial IDTitleTutorialVideo
Tutorial
Revision
Notes
Revision
Questions
6.4Factorising


In these revision notes for Factorising, we cover the following key points:

  • What is factorising?
  • What is the relationship between factorising and expanding?
  • What do all terms of an algebraic expression have in common if the expression can be factorised?
  • How to factorise an algebraic expression.
  • What do the eight special algebraic expressions relate to in factorisation?
  • What are quadratics? How to factorise them?

Factorising Revision Notes

By definition, factorising is the inverse of expanding, consisting in the use of common factors to write an expression as a product of factors.

The simplest factorisation scheme involves three quantities, where one of them is a factor of the other two. In symbols, we write it as

ab ± ac = a ∙ (b ± c)

The common term is factorised while the rest of each term of the expression is written in brackets multiplied by the factorised part. The signs of terms in brackets do not change unless the common factor is positive or of unknown sign; otherwise (if the common factor is negative) all terms in brackets change sign.

An expression is fully factorised when you don't have more operations to do in the main expression in brackets after the common factor is taken out.

The eight special algebraic identities are examples of factorisation when considered in the reverse way.

Quadratics are algebraic expressions of the form

ax2 + bx + c

where a, b and c are numbers (a and b are coefficients and c is a constant), while x is the only variable of such expressions. If such quadratics are factorised, the expression will be written in the form

(x - m)(x - n)

where m and n are two numbers such that when the factorised quadratics expands, we can obtain the original form of this expression. When expanding the last expression, we obtain

x2 - (m + n) ∙ x + mn

In this way, if the two conditions

-(m + n) = b

and

mn = c

meet simultaneously in a given quadratic expression, then it is possible to factorise it.

The factorisation of quadratics of the form ax2 + bx + c where a ≠ 0 involves an extra step which consists of finding another combination of numbers that gives the coefficient a in addition to the two combinations used in the case when a = 1. In this way, we have to find the other two numbers p and q which must meet the condition

p ∙ q = a

To factorise such quadratics, we have three conditions: one for each coefficient (constant). They are:

p ∙ q = a
p ∙ q ∙ n + m = -b
m ∙ n = c

In q or p is 1 or -1, we must write the given quadratics in the form

(ax - m)(x - n)

or

(pqx - m)(x - n)

However, if we are not sure whether q or p are 1 or -1, we apply a more generic approach by using the formula

(px - m)(qx - n)

which gives the new expression for the coefficient b:

-np - mq = b

while the rest of coefficients are found by the same method as before.

Whats next?

Enjoy the "Factorising" revision notes? People who liked the "Factorising" revision notes found the following resources useful:

  1. Revision Notes Feedback. Helps other - Leave a rating for this revision notes (see below)
  2. Expressions Math tutorial: Factorising. Read the Factorising math tutorial and build your math knowledge of Expressions
  3. Expressions Video tutorial: Factorising. Watch or listen to the Factorising video tutorial, a useful way to help you revise when travelling to and from school/college
  4. Expressions Practice Questions: Factorising. Test and improve your knowledge of Factorising with example questins and answers
  5. Check your calculations for Expressions questions with our excellent Expressions calculators which contain full equations and calculations clearly displayed line by line. See the Expressions Calculators by iCalculator™ below.
  6. Continuing learning expressions - read our next math tutorial: Algebraic Fractions

Help others Learning Math just like you

Please provide a rating, it takes seconds and helps us to keep this resource free for all to use

[ 6 Votes ]

We hope you found this Math tutorial "Factorising" useful. If you did it would be great if you could spare the time to rate this math tutorial (simply click on the number of stars that match your assessment of this math learning aide) and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of math and other disciplines.

Expressions Calculators by iCalculator™