Special product as identities

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Special product as identities
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• Identify constants ,variables and term from the
  following-
• 2, a ,4 ,b ,6,-3,0,c,x , 3ab 2 a b , y ,2 x y
• constants are- 2,4,6,-3,0
• (All counting numbers having fixed value are
  called constants)
• a, b, c, x, y
• (All English alphabets which do not have fixed
  values are called variables)
• . Terms are-3ab 2 a b ,2 x y
• (When constants and variables are connected
  with the help of sign x and is called a
  term.)
• What do you mean by algebraic expression ?
• When terms are connected with the help of sign
  and - is called an algebraic expression
• Tell the types of algebraic expression ?
• There are four types of algebraic expression
• 1 ) monomial expression
• 2) binomial expression
• 3 ) trinomial expression
• 4) polynomial expression

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• Identify the monomial, binomial, trinomial and
  polynomial-
• 2ab ,3a-4b6c , 7x2,9a-3ab3abc-2bc
• Monomial - 2ab
• Binomial - 7x2
• Trinomial - 3a-4b6c
• Multinomial - 9a-3ab3abc-2bc
• What is the coefficient of a in 8abc and 2a
• (Any factors of a term is called a co-efficient)
• coefficient of a in 8 abc is 8bc and in 2a is 2
• What is the degree of 4x2 -2x and 4y-2 ?
• (The highest power of the variable is known as
  degree)
• Degree of 4x2 -2x is 2
• and 4y-2 is 1

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Rules of plus and minus

• , means plus
• Eg 7y 2y 9y
• 19x 2x 21x
• -,- means plus ( sign of minus and add the
  number)
• Eg -9f - 6f -15f
• -19ab - 2ab -21ab
• ,- or -, put the sign of bigger term and
  subtract the number.
• Eg 7xy - 9xy -2xy
• -5xyz 9xyz 4xyz
• -67bc 2bc -65bc

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Related examples

• Add 9 a ,7 a
• Sol - 9 a
• 7 a
• 16 a
• Add 5 b ,- 3 b
• Sol - - 5b
• -3b
• - 8 b

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Practice questions

• 2a2-6a2
• 4ab3ab
• -5x3x
• 2x5x-10x
• -5xy-10xy
• 22p10p
• -abc9abc
• -10x2y2x2y

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Rules of multiplication and division

• x put the plus sign and multiply the
  number
• - x put the minus sign and multiply the
  number
• x - or x (unlike signs) put the
  minus sign multiply the numbers.
• Note - same rules are applicable for division

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Some examples related to multiplication and
division

• 6c x -3a -18 ac
• -2a x 3 b -6 ab
• -5a x -5b 25 ab
• 11z x 2 a 22 za
• 16ab 4
• 4 ab
• -10 ab -5 a
• 2 b
• -20 xy 2 y
• -10 x
• 5 a - 1
• -5

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• Multiply (-3 ab) (2 b) (- 4 ab)
• Here we multiply constant and variable separately
  
• ( - x x - ) (3 x 2 x 4 ) ( a x b x b x a x
  b)
• 24 a2 b3

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• Divide 14 ab 7 b2 by 7a
• Sol - 14 ab 7 b2
• - 7a
• Step 1- separate the terms
• 14 a b 7 b2
• -7a - 7a
• Step 2- apply rule of division
• -2 b b2
• a

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• Find the square of 2x
• ( 2x )2
• (2x) x (2x)
• 4 x11
• 4 x2

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• Find the square of ab
• Sol - (ab )2
• a x a x b x b
• a11 b1 1
• a2 b2

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• Find the square root of 36 a2
• sol - v 36 a2
• v 6 x 6 x a x a
• 6 a

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To find the product of 2 binomials type 1

• Find the product of (x 1) (x4)
• P . k testing-
• 1)What is given?
• Two terms
• 2)Which is the first term?
• (x1)
• 2)Which is the second term?
• ( x4)
• 3)What type of expression is it?
• Binomial
• 4)Which sign is in between these to binomial?
• Multiply
• 5)Rule to multiply
• (x 1) (x4)
• x(x4) 1(x4)
• x 24x x 4
• x 2 5x 4

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To find the product of 2 binomials

• Find the product of (x -2) (x4)
• P . k testing-
• 1)What is given?
• Two terms
• 2)Which is the first term?
• (x-2)
• 2)Which is the second term?
• ( x4)
• 3)What type of expression is it?
• Binomial
• 4)Which sign is in between these to binomial?
• Multiply
• 5)Rule to multiply
• (x -2) (x4)
• x(x4) - 2(x4)
• x 24x -x -8
• x 2 3x -8

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To find the product of 2 binomials

• Find the product of (x -2) (x-6)
• P . k testing-
• 1)What is given?
• Two terms
• 2)Which is the first term?
• (x-2)
• 2)Which is the second term?
• ( x-6)
• 3)What type of expression is it?
• Binomial
• 4)Which sign is in between these to binomial?
• Multiply
• 5)Rule to multiply
• (x -2) (x-6)
• x(x-6) - 2(x - 6)
• x 2-6x -2x 12
• x 2 -8x 12

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practice questions

• Find the product of following
• (p8) (p 3)
• (x20)(x5)
• (x-10)(x3)
• (y-7)(y2)
• (z-2)(z-4)
• (a20)(a-2)

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Type 2

• Multiply (3x 2) ( 4x -7)
• Sol-
• (3x 2) ( 4x -7)
• 3x(4x-7) 2(4x-7)
• 12 x2 -21x 8 x -14
• 12 x 2-13x -14

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practice questions

• (2x 5) (3x 2)
• (3x 3)( 4x -4)
• (4z3)(6z -2)
• (3y-2)(2y 3)
• (5s-9)(3s-2)
• (2n-0.4)(3n-0.5)
• (3m - 1 )( 2m - 1 )
• 2 3

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Type 3 To find square of binomial expression

• Expand (5b-6c)2
• What is given?
• A binomial expression
• What is the power of this binomial expression?
• two
• What does it mean?
• It means that we have to find the square of this
  binomial
• Which are the two terms of given binomials?
• 5b,6c
• Are the two terms same?
• no
• Formula used-
• (a-b) 2a-2ab b2
• (ab) 2 a22ab b2
• Solution-
• then let a5b and b6c
• It is of the form
• (a-b) 2 a2-2ab b2

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Related questions

• Solve
• (ax by) 2
• (x-6) 2
• (2a-7b) 2
• (2a7b)2

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To express a data as a perfect square

• Express x 2 14x 49 as perfect square
• What is perfect square of 64
• 8
• What is perfect square of 49 b 2
• 7b
• What is perfect square of x 2
• X
• what is perfect square of 121 x 2 y 2
• 11xy
• Solution- x 2 14x 49
• (x )2 14x (70) 2
• it is in the form
• a 2 2ab b 2 (a b) 2
• here a x
• b 7
• (x7) 2

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Related questions

• Express as a perfect square
• X2-14x49
• 4-20a25a2
• 2540ab16a2b2
• 1-6x9x2

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Find the product of two binomials whose first and
second term are same but sign between them are
different

• Formula used
• (a b)( a-b) a2-b2

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• Find the product of (x6)(x-6)
• It is of the form
• (a b)( a-b) a2-b2
• Here
• a x
• b 6
• x 2- 62
• x2 - 36

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To find the term to be added to make expression
as a perfect square

• Type 1
• To convert a2b2 to a perfect square add 2ab to
  it

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• What should be added to 9b2 16c2 to
  make it a perfect square
• Solution - 9b2 16c2
• (3b) 2 (4c) 2
• Here a 3b ,b4c
• To make it perfect square,
• add ( 2x3bx4c)
• ( 24bc)
• Perfect square so obtained are
• 9b2 16c2 24bc 9b2 16c2 -24 bc
• (3b 4c) 2 ( 3b-4c) 2

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Related questions

• What should be added to 36 x 2 49 y 2
• to make it a perfect square?
• What should be added to 25 x2 64 y2
• to make it a perfect square?
• What should be added to 121y2 -100 x2
• to make it a perfect square?
• What should be added to16 x 2 36y2 to make it a
  perfect square?

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How to convert (a2 2ab) or (a2-2ab) to a perfect
square

• Hint -
• To convert (a2 2ab) or (a2 -2ab) to a perfect
  square add b2 to it.

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question

• What should be added to (a2 -14ab) to make it a
  perfect square.
• Solution- a2 -14ab
• a2 -2xax7b
• to make it perfect square ,we
  must add (7b) 2 to it
• 49 b2
• .'. The new expression is
• a2 -2xax7b 49 b2
• a2 -2xax7b (7 b)2

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Related questionstype1

• What should be added to the following to make it
  a perfect square?
• x2x
• a214ab
• 16m-24mn
• 25x220xy
• y2-y
• 100x260xy

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Type 2

• If( x1) 2 ,find the value of( x21 )
• x
  x2
• given- ( x1) 2
• x
• To find - ( x21 )
• x2
• solution-( x1) 2
• x
• Squaring both sides
• ( x1) 2 2 2
• x
• (x) 21 2(x)(1) 4
• x 2 x
• (x) 21 2 4
• x 2
• (x) 21 4-2
• x 2

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Practice questions

• If( a1) 2 ,find the value of( a21 )
• a
  a2
• If( a-1) 2 ,find the value of( a21 )
• a
  a2
• If( z1) 2 ,find the value of( z21 )
• z
  z2
• If( z-1) 2 ,find the value of( z21 )
• z
  z2

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factorization

• The factorisation of an algebraic expression
  means to express it as the product of monomials
  and the smallest degree polynomial
• H.C.F of monomial (H.C.F of numerical
  coefficient) x ( H.C.F of literal coefficient)

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How to find of monomials H.C.F

• Find the H.C.F of 4 a2b,6ab2,8a2b2
• Sol- 4 a2b 2x2 x a x ax b
• 6ab2 2x3x ax b x b
• 8a2b2 2x2x2xax a x b x b
• H.C.F of 4 a2b,6ab2,8a2b2 2xaxy
• 2ab

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Practice questions

• How to find H.C.F of following monomials
• 3x , 6x
• 12x2y ,16xy2
• 15pq ,20 q r ,25 r p
• 3x,6y,9z
• 30a2b2c2 , -18a2b c 2,6 abc2
• 2x ,4xy

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How to factorise the given expression when a
monomial is the common factor of all the terms

• Step 1- find by inspection the greatest
  monomial by which each term of the given
  expression can be divided .
• Step 2- divide each term by this monomial
  .enclose the quotients within a bracket and keep
  the common monomial outside the bracket.

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• Factorise 25 a2 b 35 a b2
• sol- 25 a2 b 35 a b2
• 5x5xa x a x b 7x5 xax b x b
• 5ab (5a7b)

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Practice questions

• x2x
• 9a 2-6ax
• 20m-25n15p
• 9-27p236p
• 12abc23ab2c

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How to factorise when the given algebraic
expression has a common binomial or trinomial

• Rule - take out the common binomial or trinomial
  as a multiple and divide throughout by this
  common factor

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Factorise x(x4) 3(x4)

• Sol- x(x4) 3(x4)
• (x4)(x3)

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Practice questions

• y(x3)7(x3)
• 3a(xy)-7b(xy)
• 2y(xy)3(xy)
• 3x(x-4)-6y(x-4)

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How to factorise when a grouping gives rise to
common factors

• Step 1- arrange the terms of the given
  expression in groups in such a way that each
  group has the same common factor
• Step 2- factorise each group.
• Step 3- take out the factor which is common in
  each group.

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Factorise axbxayby

• solution-
• axbxayby
• x (a b) y (a b)
• (x y) (a b)

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Practice questions

• ax bx ac bc
• x2 -ax-bx ab
• x2-ax-bxab
• 6ab-b2 12 ac -2bc

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How to factorise when the given expression is
expressible as the difference of two squares

• Rule- use formula
• (a2-b2) (a b) (a-b)

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Factorise 4z2 -49

• 4z2 -49
• (2z)2 (7)2
• it is of the form
• (a2-b2) (a b) (a-b)
• Here a 2z
• b7
• (2z7)(2z-7)

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Practice questions

• X2-9
• 4y2-1
• a4 - b4
• m2-121
• 100a2 121y2
• 36 -z2

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Mental math's questions

• Find the product of
• (x2)(x5)
• (p8)(p3)
• ( x-4)(x3)
• (a0.2)(a0.7)
• (2x5)(2x8)
• (m-11)(m-4)
• (5x-7)(5x3)

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• Expand
• (a3b) 2
• (5xy) 2
• (4x7y)2
• (3x2y) 2
• 4 9

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• Expand
• (p-3q) (a3b) 2
• (5xy) 2
• (4x7y)2
• (3x2y) 2
• 4 9
• (5x-y) 2
• (3x-7y)2
• (2x -3y) 2
• 5 4

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Find the product

• (2x) (2-x)
• (p 2 q 2)(p 2 q 2)
• (2x) (2-x)
• (2x -3y) (2x 3y)
• 3 4 3 4

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Identify monomial ,binomial ,trinomial

• 25 a2 b 35 a b2
• 36 a2
• 4x2 -7y 8z
• 36 a2
• 5pq
• 3x 5y z

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Find H.C.F of

• 6x2 ,8 x y
• 12x2y ,16xy2
• 30abc ,-18 a b c
• 4m2, 6m, 8mn2

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factorise

• m2 121
• 4y2 -1
• 9p2 -q2
• 81 49 x2
• x2 - 16y2
• ax by acbc
• m2 mn4m-4n