# Products and factors

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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.

$\begin{array}{c}\hfill a·{x}^{k}+b·x+{c}^{m}=0\\ \hfill d·{y}^{p}+e·y+f\le 0\end{array}$
 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 $d·{y}^{p}+e·y+f\le 0$ 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. $\left(ax+b\right)$ and $\left(cx+d\right)$ . If these two binomials are multiplied, the following is the result:

$\begin{array}{ccc}\hfill \left(a·x+b\right)\left(c·x+d\right)& =& \left(ax\right)\left(c·x+d\right)+b\left(c·x+d\right)\hfill \\ & =& \left(ax\right)\left(cx\right)+\left(ax\right)d+b\left(cx\right)+b·d\hfill \\ & =& a{x}^{2}+x\left(ad+bc\right)+bd\hfill \end{array}$

Find the product of $\left(3x-2\right)\left(5x+8\right)$

1. $\begin{array}{ccc}\hfill \left(3x-2\right)\left(5x+8\right)& =& \left(3x\right)\left(5x\right)+\left(3x\right)\left(8\right)+\left(-2\right)\left(5x\right)+\left(-2\right)\left(8\right)\hfill \\ & =& 15{x}^{2}+24x-10x-16\hfill \\ & =& 15{x}^{2}+14x-16\hfill \end{array}$

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

${\left(ax+b\right)}^{2}={a}^{2}{x}^{2}+2abx+{b}^{2}$

If the two terms are $ax+b$ and $ax-b$ then their product is:

$\left(ax+b\right)\left(ax-b\right)={a}^{2}{x}^{2}-{b}^{2}$

This is known as the difference of two squares .

## Factorisation

Factorisation is the opposite of expanding brackets. For example expanding brackets would require $2\left(x+1\right)$ to be written as $2x+2$ . Factorisation would be to start with $2x+2$ and to end up with $2\left(x+1\right)$ . 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, $2x-6{x}^{2}$ can be factorised as follows:

$2x-6{x}^{2}=2x\left(1-3x\right)$

## Investigation : common factors

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

 (a) $6y;18x$ (b) $12mn;8n$ (c) $3st;4su$ (d) $18kl;9kp$ (e) $abc;ac$ (f) $2xy;4xyz$ (g) $3uv;6u$ (h) $9xy;15xz$ (i) $24xyz;16yz$ (j) $3m;45n$

## Difference of two squares

We have seen that:

$\left(ax+b\right)\left(ax-b\right)={a}^{2}{x}^{2}-{b}^{2}$

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

$\left(ax+b\right)\left(ax-b\right)$

Therefore,

${a}^{2}{x}^{2}-{b}^{2}=\left(ax+b\right)\left(ax-b\right)$

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

Factorise completely: ${b}^{2}{y}^{5}-3ab{y}^{3}$

1. $\begin{array}{ccc}\hfill {b}^{2}{y}^{5}-3ab{y}^{3}& =& b{y}^{3}\left(b{y}^{2}-3a\right)\hfill \end{array}$

Factorise completely: $3a\left(a-4\right)-7\left(a-4\right)$

1. $\left(a-4\right)$ is the common factor
$\begin{array}{ccc}\hfill 3a\left(a-4\right)-7\left(a-4\right)& =& \left(a-4\right)\left(3a-7\right)\hfill \end{array}$

Factorise $5\left(a-2\right)-b\left(2-a\right)$

1. $\begin{array}{ccc}\hfill 5\left(a-2\right)-b\left(2-a\right)& =& 5\left(a-2\right)-\left[-b\left(a-2\right)\right]\hfill \\ & =& 5\left(a-2\right)+b\left(a-2\right)\hfill \\ & =& \left(a-2\right)\left(5+b\right)\hfill \end{array}$

## Recap

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

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

3. Factorise completely:
 (a) $7a+4$ (b) $20a-10$ (c) $18ab-3bc$ (d) $12kj+18kq$ (e) $16{k}^{2}-4k$ (f) $3{a}^{2}+6a-18$ (g) $-6a-24$ (h) $-2ab-8a$ (i) $24kj-16{k}^{2}j$ (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) $4\left(y-3\right)+k\left(3-y\right)$ (n) $a\left(a-1\right)-5\left(a-1\right)$ (o) $bm\left(b+4\right)-6m\left(b+4\right)$ (p) ${a}^{2}\left(a+7\right)+a\left(a+7\right)$ (q) $3b\left(b-4\right)-7\left(4-b\right)$ (r) ${a}^{2}{b}^{2}{c}^{2}-1$

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
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Stoney Reply
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Adin Reply
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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absolutely yes
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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
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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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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
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CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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s.
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for screen printed electrodes ?
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s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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