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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 Z . If we factorise this expression we get: n ( n + 1 ) . 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 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 ( n + 1 ) ( n - 1 ) . 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 Z .


  • 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 ( a x + b ) ( a x - b )
  • 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 ( ( A + B ) ( C + D + E ) = A ( C + D + E ) + B ( C + D + E ) ) helps us to multiply a binomial and a trinomial.
  • The sum of cubes is: ( x + y ) ( x 2 - x y + y 2 ) = x 3 + y 3 and the difference of cubes is: x 3 - y 3 = ( x - y ) ( x 2 + x y + y 2 )
  • 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 - ( 1 / 9 )
    6. 5 - 5 a 2 b 6
    7. 16 b a 4 - 81 b
    8. a 2 - 10 a + 25
    9. 16 b 2 + 56 b + 49
    10. 2 a 2 - 12 a b + 18 b 2
    11. - 4 b 2 - 144 b 8 + 48 b 5
  2. Factorise completely:
    1. ( 16 x 4 )
    2. 7x 2 14x + 7xy 14y
    3. y 2 7y 30
    4. 1 x x 2 + x 3
    5. 3 ( 1 p 2 ) + p + 1
  3. Simplify the following:
    1. ( a - 2 ) 2 - a ( a + 4 )
    2. ( 5 a - 4 b ) ( 25 a 2 + 20 ab + 16 b 2 )
    3. ( 2 m - 3 ) ( 4 m 2 + 9 ) ( 2 m + 3 )
    4. ( a + 2 b - c ) ( a + 2 b + c )
  4. Simplify the following:
    1. p 2 - q 2 p ÷ p + q p 2 - pq
    2. 2 x + x 2 - 2 x 3
  5. Show that ( 2 x - 1 ) 2 - ( x - 3 ) 2 can be simplified to ( x + 2 ) ( 3 x - 4 )

  6. What must be added to x 2 - x + 4 to make it equal to ( x + 2 ) 2

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
is it a question of log
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Maths grade 10 rought draft. OpenStax CNX. Sep 29, 2011 Download for free at http://cnx.org/content/col11363/1.1
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