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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 variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
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
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Damian Reply
absolutely yes
Daniel
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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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