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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 .

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 ( 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

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+_
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
y=10×
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
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Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
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J, combine like terms 7x-4y
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im not good at math so would this help me
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f(x)= 2|x+5| find f(-6)
Prince Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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what is nanomaterials​ and their applications of sensors.
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what is nano technology
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AMJAD
preparation of nanomaterial
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
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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.
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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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