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Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
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6.4 | Factorising |
In these revision notes for Factorising, we cover the following key points:
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
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
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
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
In this way, if the two conditions
and
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
To factorise such quadratics, we have three conditions: one for each coefficient (constant). They are:
In q or p is 1 or -1, we must write the given quadratics in the form
or
However, if we are not sure whether q or p are 1 or -1, we apply a more generic approach by using the formula
which gives the new expression for the coefficient b:
while the rest of coefficients are found by the same method as before.
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