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Introduction

In this chapter you will learn how to work with algebraic expressions. You will recap some of the work on factorisation and multiplying out expressions that you learnt in earlier grades. This work will then be extended upon for Grade 10.

Recap of earlier work

The following should be familiar. Examples are given as reminders.

Parts of an expression

Mathematical expressions are just like sentences and their parts have special names. You should be familiar with the following names used to describe the parts of a mathematical expression.

a · x k + b · x + c m = 0 d · y p + e · y + f 0
Name Examples (separated by commas)
term a · x k , b · x , c m , d · y p , e · y , f
expression a · x k + b · x + c m , d · y p + e · y + f
coefficient a , b , d , e
exponent (or index) k , p
base x , y , c
constant a , b , c , d , e , f
variable x , y
equation a · x k + b · x + c m = 0
inequality d · y p + e · y + f 0
binomial expression with two terms
trinomial expression with three terms

Product of two binomials

A binomial is a mathematical expression with two terms, e.g. ( a x + b ) and ( c x + d ) . If these two binomials are multiplied, the following is the result:

( a · x + b ) ( c · x + d ) = ( a x ) ( c · x + d ) + b ( c · x + d ) = ( a x ) ( c x ) + ( a x ) d + b ( c x ) + b · d = a x 2 + x ( a d + b c ) + b d

Find the product of ( 3 x - 2 ) ( 5 x + 8 )

  1. ( 3 x - 2 ) ( 5 x + 8 ) = ( 3 x ) ( 5 x ) + ( 3 x ) ( 8 ) + ( - 2 ) ( 5 x ) + ( - 2 ) ( 8 ) = 15 x 2 + 24 x - 10 x - 16 = 15 x 2 + 14 x - 16

The product of two identical binomials is known as the square of the binomial and is written as:

( a x + b ) 2 = a 2 x 2 + 2 a b x + b 2

If the two terms are a x + b and a x - b then their product is:

( a x + b ) ( a x - b ) = a 2 x 2 - b 2

This is known as the difference of two squares .

Factorisation

Factorisation is the opposite of expanding brackets. For example expanding brackets would require 2 ( x + 1 ) to be written as 2 x + 2 . Factorisation would be to start with 2 x + 2 and to end up with 2 ( x + 1 ) . In previous grades, you factorised based on common factors and on difference of squares.

Common factors

Factorising based on common factors relies on there being common factors between your terms. For example, 2 x - 6 x 2 can be factorised as follows:

2 x - 6 x 2 = 2 x ( 1 - 3 x )

Investigation : common factors

Find the highest common factors of the following pairs of terms:

(a) 6 y ; 18 x (b) 12 m n ; 8 n (c) 3 s t ; 4 s u (d) 18 k l ; 9 k p (e) a b c ; a c
(f) 2 x y ; 4 x y z (g) 3 u v ; 6 u (h) 9 x y ; 15 x z (i) 24 x y z ; 16 y z (j) 3 m ; 45 n

Difference of two squares

We have seen that:

( a x + b ) ( a x - b ) = a 2 x 2 - b 2

Since [link] is an equation, both sides are always equal. This means that an expression of the form:

a 2 x 2 - b 2

can be factorised to

( a x + b ) ( a x - b )

Therefore,

a 2 x 2 - b 2 = ( a x + b ) ( a x - b )

For example, x 2 - 16 can be written as ( x 2 - 4 2 ) which is a difference of two squares. Therefore, the factors of x 2 - 16 are ( x - 4 ) and ( x + 4 ) .

Factorise completely: b 2 y 5 - 3 a b y 3

  1. b 2 y 5 - 3 a b y 3 = b y 3 ( b y 2 - 3 a )

Factorise completely: 3 a ( a - 4 ) - 7 ( a - 4 )


  1. ( a - 4 ) is the common factor
    3 a ( a - 4 ) - 7 ( a - 4 ) = ( a - 4 ) ( 3 a - 7 )

Factorise 5 ( a - 2 ) - b ( 2 - a )

  1. 5 ( a - 2 ) - b ( 2 - a ) = 5 ( a - 2 ) - [ - b ( a - 2 ) ] = 5 ( a - 2 ) + b ( a - 2 ) = ( a - 2 ) ( 5 + b )

Recap

  1. Find the products of:
    (a) 2 y ( y + 4 ) (b) ( y + 5 ) ( y + 2 ) (c) ( y + 2 ) ( 2 y + 1 )
    (d) ( y + 8 ) ( y + 4 ) (e) ( 2 y + 9 ) ( 3 y + 1 ) (f) ( 3 y - 2 ) ( y + 6 )


  2. Factorise:
    1. 2 l + 2 w
    2. 12 x + 32 y
    3. 6 x 2 + 2 x + 10 x 3
    4. 2 x y 2 + x y 2 z + 3 x y
    5. - 2 a b 2 - 4 a 2 b


  3. Factorise completely:
    (a) 7 a + 4 (b) 20 a - 10 (c) 18 a b - 3 b c
    (d) 12 k j + 18 k q (e) 16 k 2 - 4 k (f) 3 a 2 + 6 a - 18
    (g) - 6 a - 24 (h) - 2 a b - 8 a (i) 24 k j - 16 k 2 j
    (j) - a 2 b - b 2 a (k) 12 k 2 j + 24 k 2 j 2 (l) 72 b 2 q - 18 b 3 q 2
    (m) 4 ( y - 3 ) + k ( 3 - y ) (n) a ( a - 1 ) - 5 ( a - 1 ) (o) b m ( b + 4 ) - 6 m ( b + 4 )
    (p) a 2 ( a + 7 ) + a ( a + 7 ) (q) 3 b ( b - 4 ) - 7 ( 4 - b ) (r) a 2 b 2 c 2 - 1


Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Source:  OpenStax, Maths test. OpenStax CNX. Feb 09, 2011 Download for free at http://cnx.org/content/col11236/1.2
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