# 1.6 Fractions  (Page 2/2)

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 Odd number Even number Odd number Odd Even Even number Even Even

If we take three consecutive numbers and multiply them together, the resulting number is always divisible by three. This should be obvious since if we have any three consecutive numbers, one of them will be divisible by 3.

Now we are ready to demonstrate that ${n}^{2}+n$ is even for all $n\in {Z}$ . If we factorise this expression we get: $n\left(n+1\right)$ . If $n$ is even, than $n+1$ is odd. If $n$ is odd, than $n+1$ is even. Since we know that if we multiply an even number with an odd number or an odd number with an even number, we get an even number, we have demonstrated that ${n}^{2}+n$ is always even. Try this for a few values of $n$ and you should find that this is true.

To demonstrate that ${n}^{3}-n$ is divisible by 6 for all $n\in {Z}$ , we first note that the factors of 6 are 3 and 2. So if we show that ${n}^{3}-n$ is divisible by both 3 and 2, then we have shown that it is also divisible by 6! If we factorise this expression we get: $n\left(n+1\right)\left(n-1\right)$ . Now we note that we are multiplying three consecutive numbers together (we are taking $n$ and then adding 1 or subtracting 1. This gives us the two numbers on either side of $n$ .) For example, if $n=4$ , then $n+1=5$ and $n-1=3$ . But we know that when we multiply three consecutive numbers together, the resulting number is always divisible by 3. So we have demonstrated that ${n}^{3}-n$ is always divisible by 3. To demonstrate that it is also divisible by 2, we can also show that it is even. We have shown that ${n}^{2}+n$ is always even. So now we recall what we said about multiplying even and odd numbers. Since one number is always even and the other can be either even or odd, the result of multiplying these numbers together is always even. And so we have demonstrated that ${n}^{3}-n$ is divisible by 6 for all $n\in {Z}$ .

## Summary

• A binomial is a mathematical expression with two terms. The product of two identical binomials is known as the square of the binomial. The difference of two squares is when we multiply $\left(ax+b\right)\left(ax-b\right)$
• Factorising is the opposite of expanding the brackets. You can use common factors or the difference of two squares to help you factorise expressions.
• The distributive law ( $\left(A+B\right)\left(C+D+E\right)=A\left(C+D+E\right)+B\left(C+D+E\right)$ ) helps us to multiply a binomial and a trinomial.
• The sum of cubes is: $\left(x+y\right)\left({x}^{2}-xy+{y}^{2}\right)={x}^{3}+{y}^{3}$ and the difference of cubes is: ${x}^{3}-{y}^{3}=\left(x-y\right)\left({x}^{2}+xy+{y}^{2}\right)$
• To factorise a quadratic we find the two binomials that were multiplied together to give the quadratic.
• We can also factorise a quadratic by grouping. This is where we find a common factor in the quadratic and take it out and then see what is left over.
• We can simplify fractions by using the methods we have learnt to factorise expressions.
• Fractions can be added or subtracted. To do this the denominators of each fraction must be the same.

## End of chapter exercises

1. Factorise:
1. ${a}^{2}-9$
2. ${m}^{2}-36$
3. $9{b}^{2}-81$
4. $16{b}^{6}-25{a}^{2}$
5. ${m}^{2}-\left(1/9\right)$
6. $5-5{a}^{2}{b}^{6}$
7. $16b{a}^{4}-81b$
8. ${a}^{2}-10a+25$
9. $16{b}^{2}+56b+49$
10. $2{a}^{2}-12ab+18{b}^{2}$
11. $-4{b}^{2}-144{b}^{8}+48{b}^{5}$
2. Factorise completely:
1. $\left(16-{x}^{4}\right)$
2. ${7x}^{2}-14x+7xy-14y$
3. ${y}^{2}-7y-30$
4. $1-x-{x}^{2}+{x}^{3}$
5. $-3\left(1-{p}^{2}\right)+p+1$
3. Simplify the following:
1. ${\left(a-2\right)}^{2}-a\left(a+4\right)$
2. $\left(5a-4b\right)\left(25{a}^{2}+20\mathrm{ab}+16{b}^{2}\right)$
3. $\left(2m-3\right)\left(4{m}^{2}+9\right)\left(2m+3\right)$
4. $\left(a+2b-c\right)\left(a+2b+c\right)$
4. Simplify the following:
1. $\frac{{p}^{2}-{q}^{2}}{p}÷\frac{p+q}{{p}^{2}-\mathrm{pq}}$
2. $\frac{2}{x}+\frac{x}{2}-\frac{2x}{3}$
5. Show that ${\left(2x-1\right)}^{2}-{\left(x-3\right)}^{2}$ can be simplified to $\left(x+2\right)\left(3x-4\right)$

6. What must be added to ${x}^{2}-x+4$ to make it equal to ${\left(x+2\right)}^{2}$

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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