Understanding Engineering Mathematics

Another important example is aquadratic expression inxsuch as 2x^2 + 7 x−4, of
general form

ax^2 +bx+c

wherea,b,care again ‘constants’, independent ofx.
In order to use algebra for more advanced topics such as factorisation or partial fractions,
we need to be able to perform multiplication of simple algebraic expressions quickly and
accurately. The key to this is dealing with multiplication of bracket expressions such as
(a+b)(c+d). All you need for this is to know that the expressiona(c+d)is ‘expanded’
by multiplyingcanddbya:

a(c+d)=ac+ad

This is called thedistributive rule: we say ‘multiplication distributes over addition’.
Real numbers also satisfy theassociative rulea(bc)=(ab)cand thecommutative rule
ab=ba. Another useful property of real numbers is that ifab=0 then one or both ofa,
bmust be zero. Note that these statements arerulesof algebra, which will not necessarily
hold for algebras not based on the real numbers – for example the commutative law does
not hold in matrix algebra, the subject of Chapter 13. However, given these rules we can
use them to handle more complicated expressions. For example, we can now expand out
any number of bracket expressions.

Examples
(a+b)(c+d)=(a+b)c+(a+b)d
=ac+bc+ad+bd
( 3 x+ 2 )(x− 1 )=( 3 x+ 2 )x+( 3 x+ 2 )(− 1 )
= 3 x^2 + 2 x− 3 x− 2
= 3 x^2 −x− 2

Note the collection of the ‘like’ terms, 2x− 3 x=−x. You should always tidy up
calculations in this way.
Always be careful with the treatment of signs and brackets, particularly when they are
mixed. If in doubt leave minus signs bracketed:

(− 1 )(x− 1 )=(− 1 )x−(− 1 ) 1

=−x+ 1

Never omit brackets out of laziness.
Later you will need toreversesome of the above operations. For example, you could
be givenac+bc+ad+bdand asked to ‘factorise’ it – that is, convert it to the form
(a+b)(c+d). This is much easier to do if you are highly proficient at multiplying such
expressions in the first place – to do something ‘standing on your head’ it helps if you can
do it the right way up first!! There is only one way to achieve this proficiency – practice,
so try the reinforcement exercises until you are confident.