# 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 ${\left(x+2\right)}^{2}$

what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
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
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?
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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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