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Special product as identities

Special product as identities. 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

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Special product as identities

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  1. Special product as identities

  2. 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

  3. Identify the monomial, binomial, trinomial and polynomial:- 2ab ,3a-4b+6c , 7x+2,9a-3ab+3abc-2bc Monomial :- 2ab Binomial :- 7x+2 Trinomial :- 3a-4b+6c Multinomial :- 9a-3ab+3abc-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

  4. 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

  5. Related examples • Add + 9 a ,+7 a Sol :- +9 a +7 a +16 a Add – 5 b ,- 3 b Sol :- - 5b -3b - 8 b

  6. Practice questions • +2a2-6a2 • +4ab+3ab • -5x+3x • 2x+5x-10x • -5xy-10xy • +22p+10p • -abc+9abc • -10x2y+2x2y

  7. 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

  8. 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

  9. 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

  10. 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

  11. Find the square of 2x • ( 2x )2 = (2x) x (2x) = 4 x1+1 = 4 x2

  12. Find the square of ab Sol :- (ab )2 = a x a x b x b = a1+1 b1 +1 = a2 b2

  13. Find the square root of 36 a2 sol :- √ 36 a2 = √ 6 x 6 x a x a = 6 a

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

  15. To find the product of 2 binomials • Find the product of (x -2) (x+4) P . k testing:- 1)What is given? Two terms 2)Which is the first term? (x-2) 2)Which is the second term? ( x+4) 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+4) = x(x+4) - 2(x+4) =x 2+4x -x -8 = x 2 +3x -8

  16. 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

  17. practice questions • Find the product of following • (p+8) (p +3) • (x+20)(x+5) • (x-10)(x+3) • (y-7)(y+2) • (z-2)(z-4) • (a+20)(a-2)

  18. 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

  19. practice questions • (2x +5) (3x +2) • (3x + 3)( 4x -4) • (4z+3)(6z -2) • (3y-2)(2y +3) • (5s-9)(3s-2) • (2n-0.4)(3n-0.5) • (3m - 1 )( 2m - 1 ) 2 3

  20. 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) 2=a-2ab +b2 (a+b) 2 =a2+2ab +b2 Solution:- then let a=5b and b=6c It is of the form (a-b) 2 =a2-2ab +b2 =(5b) 2 -2x5bx6c+(6c) 2 = 25b2 -60bc +35 c 2 Type 3 To find square of binomial expression

  21. Related questions • Solve • (ax – by) 2 • (x-6) 2 • (2a-7b) 2 • (2a+7b)2

  22. 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 =(x+7) 2

  23. Related questions • Express as a perfect square • X2-14x+49 • 4-20a+25a2 • 25+40ab+16a2b2 • 1-6x+9x2

  24. 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

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

  26. To find the term to be added to make expression as a perfect square • Type 1 • To convert a2+b2 to a perfect square add +2ab to it

  27. What should be added to 9b2 + 16c2 to make it a perfect square Solution :- 9b2 + 16c2 = (3b) 2 +(4c) 2 Here a =3b ,b=4c 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

  28. 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?

  29. 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.

  30. 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

  31. Related questionstype1 • What should be added to the following to make it a perfect square? • x2+x • a2+14ab • 16m-24mn • 25x2+20xy • y2-y • 100x2+60xy

  32. Type 2 • If( x+1) =2 ,find the value of( x2+1 ) x x2 given:- ( x+1) =2 x To find :- ( x2+1 ) x2 solution:-( x+1) =2 x Squaring both sides ( x+1) 2 =2 2 x (x) 2+1 +2(x)(1) =4 x 2 x (x) 2+1 +2 =4 x 2 (x) 2+1 =4-2 x 2 (x) 2+1 =2 x 2

  33. Practice questions • If( a+1) =2 ,find the value of( a2+1 ) a a2 • If( a-1) =2 ,find the value of( a2+1 ) a a2 • If( z+1) =2 ,find the value of( z2+1 ) z z2 • If( z-1) =2 ,find the value of( z2+1 ) z z2

  34. 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)

  35. 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

  36. 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

  37. 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.

  38. 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 (5a+7b)

  39. Practice questions • x2+x • 9a 2-6ax • 20m-25n+15p • 9-27p2+36p • 12abc2+3ab2c

  40. 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

  41. Factorise x(x+4) +3(x+4) • Sol:- x(x+4) +3(x+4) • = (x+4)(x+3)

  42. Practice questions • y(x+3)+7(x+3) • 3a(x+y)-7b(x+y) • 2y(x+y)+3(x+y) • 3x(x-4)-6y(x-4)

  43. 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.

  44. Factorise ax+bx+ay+by • solution:- ax+bx+ay+by =x (a +b) +y (a +b) =(x +y) (a +b)

  45. Practice questions • ax +bx+ ac+ bc • x2 -ax-bx+ ab • x2-ax-bx+ab • 6ab-b2 +12 ac -2bc

  46. How to factorise when the given expression is expressible as the difference of two squares • Rule:- use formula • (a2-b2) =(a+ b) (a-b)

  47. Factorise 4z2 -49 • 4z2 -49 =(2z)2 –(7)2 it is of the form • (a2-b2) =(a+ b) (a-b) Here a =2z b=7 =(2z+7)(2z-7)

  48. Practice questions • X2-9 • 4y2-1 • a4 - b4 • m2-121 • 100a2 – 121y2 • 36 -z2

  49. Mental math's questions • Find the product of • (x+2)(x+5) • (p+8)(p+3) • ( x-4)(x+3) • (a+0.2)(a+0.7) • (2x+5)(2x+8) • (m-11)(m-4) • (5x-7)(5x+3)

  50. Expand • (a+3b) 2 • (5x+y) 2 • (4x+7y)2 • (3x+2y) 2 4 9

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