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1a factorization ok

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Maninath Neupane

This document provides examples of factorizing polynomials using various factorization techniques. It contains 30 questions with step-by-step solutions demonstrating how to factorize expressions involving single variables, binomials, trinomials, and polynomials using formulas like difference of squares, perfect square trinomials, and grouping. The techniques shown include finding common factors, using factor trees, recognizing patterns, and applying factorization identities.

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Curriculum on factorization Factorization a2-b2, trinomial with perfect square, trinomial in the form of ax2 bx c. PRESENTED BY MANINATH NEUPANE CLASS 8
Factorization
Q 1.Factorise: 2ax+4ay ax ay a x a y a x a x a y a y a y a y 2a (x+2y)
Q 1.Factorise: 2ax+4ay =2 . =2.2 2ax + 4aySolution: 2ax 4ay = a . x . a . y ( + ) Rule 1. We have to take common factors as common and remaining factors should be kept inside parenthesis.
Q 2.Factorise: 4a2b-6ab2+10ab ( 4a2b-6ab2+10ab 4a2b 6ab2 10ab =2. 2. a. a. b 2. 3. a. b. b= 2. 5. a. b= - + ) Solution: = Rule 1. We have to take common factors as common and remaining factors should be kept inside parenthesis.
Q 3. Resolve into factors: 2x (a + b)- 3y (a + b) Solution: 2x (a + b)- 3y (a + b) 2x (a + b)= 3y (a + b)= 2 . x . (a+b) 3 . y . (a+b) = ( - ) Rule 1. We have to take common factors as common and remaining factors should be kept inside parenthesis.
Q 4.Factorise: a2-ax+ab-bx Solution: a2-ax + ab-bx = ( a2= a . a ax= a . x- ) + b(a - x) =(a-x)(a+b) Rule 2. We have to arrange the terms in groups such that each group has common factor.
Q 5.Factorise: a(x2-y2)+x(y2-a2) Solution: a(x2-y2)+x(y2-a2) =ax2-ay2+xy2-xa2 =ax2+xy2-ay2-xa2 =x(ax+y2)-a(y2+xa) =x(ax+y2)-a(xa+y2) =(ax+y2)(x-a) We have to simplify the expression first so that we can factorize.
Q 6.Factorise: x2-9 Solution: x 2 - 9 = x 2 3 2 x2 32 = a 2 b 2 Let x=a and 3=b = (a b)(a + b) = ( x 3 )( x + 3 ) a 2 b 2 = ( a b ) ( a + b ) Putting the value of a and b. Rule 3. We have to use the suitable formula.
Q 6.Factorise: x2-9 Solution: x 2 - 9 = x 2 3 2 = ( x 3 )( x + 3 ) a 2 b 2 = ( a b ) ( a + b ) Rule 3. We have to use the suitable formula.
Q 6.Factorise: x2-9 Solution: x 2 - 9 = x 2 3 2 = ( x 3 )( x + 3 ) a 2 b 2 = ( a b ) ( a + b ) Rule 3. We have to use the suitable formula.
Q 7.Factorise: 25x2-9y2 Solution: 25x2-9y2 a2 b2 =(ab)(a+b)= (5x)2 (3y)2 = ( 5x 3y )( 5x + 3y ) Rule 3. We have to use the suitable formula.
Q 8.Factorise: a4-16 Solution: a4-16 =(a2)2-42 =(a2-4)(a2+4) =(a2-22)(a2+4) =(a-2)(a+2)(a2+4) a2 b2 =(ab)(a+b) Rule 3. We have to use the suitable formula.
Q 9.Factorise: (x2+y2)2-x2y2 Solution: (x2+y2)2-x2y2 =(x2+y2)2-(xy)2 ={(x2+y2)-(xy)}{(x2+y2)+(xy)} =(x2+y2-xy)(x2+y2+xy) =(x2-xy+y2)(x2+xy+y2) a2 b2 =(ab)(a+b) Rule 3. We have to use the suitable formula.
Q 10.Factorise: 4-(a-b)2 Solution: 4-(a-b)2 =22-(a-b)2 ={2-(a-b)}{2+(a-b)} =(2-a+b)(2+a-b) a2 b2 =(ab)(a+b) Rule 3. We have to use the suitable formula.
Q 11. Simplify by factorization process: 722-622. Solution: 722-622 =(72-62)(72+62) =10134 =1340 We have to use the suitable formula.
Q 12. Simplify by factorization process: 10199 Solution: 10199 = (100+1)(100-1) = (10000-1) = 9999 We have to use the suitable formula.
Q 13. If a+b=8 and ab =15, find the value of a and b. numbers(a & b) sum(a+b) = 8 product(a.b) =15 3+5=8 1.7=7 Numbers are 3 &5. When a=3 then b=5 and when a=5 then b=3. 1 and 7 2 and 6 3 and 5 1+7=8 2+6=8 2.6=12 3.5=15 Solution: We use hit and trial method.
Q 14. If a+b=-10 and ab=24,find the value of a and b. numbers(a & b) a.b=24 a+b=-10 -3.-8=24 -1-24=-25 Numbers are -4 & -6. When a=-4 then b=-6 and when a=-6 then b=-4. -1 and -24 -2 and -12 -3 and -8 -1.-24=24 -2.-12=24 -2-12=-14 -3-8=-11 Solution: -4 and -6 -4.-6=24 -4-6=-10 We use hit and trial method.
Q 15. Factorize: x2+5x+6 x2 1x 1x 1x 1x 1x 1 1 1 1 1 1 x+3 x+2 x2+5x+6=(x+3)(x+2) What did we do here?
Q 15. Factorize: x2+5x+6 Solution: x2+5x+6 Rule 4. Method 1. Multiply coefficient of x2 and constant term. 16=6(product) 2. Find the possible factors of the product 6. 6=16 6=23 3. Remember the sign of constant term. = x2+(3+2)x+6 = x2+3x+2x+6 = x(x+3)+2(x+3) = (x+3)(x+2) 4. If sign of constant term is + then choose the pair of factors whose sum is 5 (coefficient of x)
Q 16. Resolve into factors: x2-9x+20 Rule 4. Method 1. Multiply coefficient of x2 and constant term. 120=20(product) 2. Find the possible factors of the product 20. 20=120 20=210 4. If sign of constant term is + then choose the pair of factors whose sum is 9 (coefficient of x) 20=45 Solution: x2-9x+20 = x2-(5+4)x+20 = x2-5x-4x+20 = x(x-5)-4(x-5) = (x-5)(x-4) 3. Remember the sign of constant term.
Q 17. Factorize: x2+3x-18 Rule 4. Method 1. Multiply coefficient of x2 and constant term. 118=18(product) 2. Find the possible factors of the product 18. 18=118 18=29 18=36 Solution: x2+3x-18 =x2+(6-3)x-18 =x2+6x-3x-18 =x(x+6)-3(x+6) =(x+6)(x-3) 4. If sign of constant term is - then choose the pair of factors whose difference is 3 (coefficient of x) 3. Remember the sign of constant term.
Q 18. Factorize: x2-5x-14 Rule 4. Method 1. Multiply coefficient of x2 and constant term. 114=14(product) 2. Find the possible factors of the product 14. 14=114 14=27 4. If sign of constant term is - then choose the pair of factors whose difference is 3 (coefficient of x) 3. Remember the sign of constant term. Solution: x2-5x-14 =x2-(7-2)x-14 =x2-7x+2x-14 =x(x-7)+2(x-7) =(x-7)(x+2)
Q 19. Factorize: 2x2+7x+3 Rule 4. Method 1. Multiply coefficient of x2 and constant term. 23=6(product) 2. Find the possible factors of the product 6. 6=16 6=23 4. If sign of constant term is + then choose the pair of factors whose sum is 7 (coefficient of x) 3. Remember the sign of constant term. Solution: 2x2+7x+3 =2x2+7x+3 =2x2+(6+1)x+3 =2x2+6x+1x+3 =2x(x+3)+1(x+3) =(x+3)(2x+1)
Q 20. Factorize: x2+x-30 Solution: x2+x-30 Rule 4. Method 1. Multiply coefficient of x2 and constant term. 130=30(product) 2. Find the possible factors of the product 30. 30=130 30=215 4. If sign of constant term is - then choose the pair of factors whose difference is 1 (coefficient of x). 3. Remember the sign of constant term. 30=310 30=56 =x2+x-30 =x2+(6-5)x-30 =x2+6x-5x-30 =x(x+6)-5(x+6) =(x+6)(x-5)
Q 21. Factorize: x2+4x+4 Solution: x2+4x+4 =x2+2.x.2+22 =(x+2)2 Rule 3. We have to use the suitable formula. Rule 4. Or,
Q 22. Factorize: 4x2-12x+9 Solution: 4x2-12x+9 = (2x)2-2.2x.3+32 = (2x-3)2 Rule 3. We have to use the suitable formula. Rule 4. Or,
Q 23. Factorize: 4x 8 Q 24. Factorize: 4x3 - 6x2 + 8x Q 25. Factorize: xm + ym + xa + ya Q 26. Factorize: a( a + 4 ) + 2 ( a + 4 ) Q 27. Factorize: a( a 3 ) - 4( 3 a ) Q 28. Factorize: a b + 1 + ab Q 29. Factorize: x2y + xy2z + zx + yz2
Q 30. Factorize: x2 1 Q 31. Factorize: 4x2 9b2 Q 32. Factorize: 1-36x2 Q 33. Factorize: x2-36 Q 34. Factorize: 16a2 25b2 Q 35. Factorize: 72x3 50x Q 36. Find the area of shaded part. y y 2 2
Q 37. Factorize: x2 + 10x +25 Q 38. Factorize: p2 24p+144 Q 39. Factorize: 9a2 66a + 121 Q 40. Factorize: x2 + 9x +18 Q 41. Factorize: x2 - 9x +18 Q 42. Factorize: x2 + 3x 18 Q 43. Factorize: x2 - 3x 18 Q 44. Factorize: 15x2 x 2 Q 45. Factorize: 125x3 + 8a3 Q 46. Factorize: (x y )3 + 8(x + y)3
1a factorization ok

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1a factorization ok

  • 1. Curriculum on factorization Factorization a2-b2, trinomial with perfect square, trinomial in the form of ax2 bx c. PRESENTED BY MANINATH NEUPANE CLASS 8
  • 3. Q 1.Factorise: 2ax+4ay ax ay a x a y a x a x a y a y a y a y 2a (x+2y)
  • 4. Q 1.Factorise: 2ax+4ay =2 . =2.2 2ax + 4aySolution: 2ax 4ay = a . x . a . y ( + ) Rule 1. We have to take common factors as common and remaining factors should be kept inside parenthesis.
  • 5. Q 2.Factorise: 4a2b-6ab2+10ab ( 4a2b-6ab2+10ab 4a2b 6ab2 10ab =2. 2. a. a. b 2. 3. a. b. b= 2. 5. a. b= - + ) Solution: = Rule 1. We have to take common factors as common and remaining factors should be kept inside parenthesis.
  • 6. Q 3. Resolve into factors: 2x (a + b)- 3y (a + b) Solution: 2x (a + b)- 3y (a + b) 2x (a + b)= 3y (a + b)= 2 . x . (a+b) 3 . y . (a+b) = ( - ) Rule 1. We have to take common factors as common and remaining factors should be kept inside parenthesis.
  • 7. Q 4.Factorise: a2-ax+ab-bx Solution: a2-ax + ab-bx = ( a2= a . a ax= a . x- ) + b(a - x) =(a-x)(a+b) Rule 2. We have to arrange the terms in groups such that each group has common factor.
  • 8. Q 5.Factorise: a(x2-y2)+x(y2-a2) Solution: a(x2-y2)+x(y2-a2) =ax2-ay2+xy2-xa2 =ax2+xy2-ay2-xa2 =x(ax+y2)-a(y2+xa) =x(ax+y2)-a(xa+y2) =(ax+y2)(x-a) We have to simplify the expression first so that we can factorize.
  • 9. Q 6.Factorise: x2-9 Solution: x 2 - 9 = x 2 3 2 x2 32 = a 2 b 2 Let x=a and 3=b = (a b)(a + b) = ( x 3 )( x + 3 ) a 2 b 2 = ( a b ) ( a + b ) Putting the value of a and b. Rule 3. We have to use the suitable formula.
  • 10. Q 6.Factorise: x2-9 Solution: x 2 - 9 = x 2 3 2 = ( x 3 )( x + 3 ) a 2 b 2 = ( a b ) ( a + b ) Rule 3. We have to use the suitable formula.
  • 11. Q 6.Factorise: x2-9 Solution: x 2 - 9 = x 2 3 2 = ( x 3 )( x + 3 ) a 2 b 2 = ( a b ) ( a + b ) Rule 3. We have to use the suitable formula.
  • 12. Q 7.Factorise: 25x2-9y2 Solution: 25x2-9y2 a2 b2 =(ab)(a+b)= (5x)2 (3y)2 = ( 5x 3y )( 5x + 3y ) Rule 3. We have to use the suitable formula.
  • 13. Q 8.Factorise: a4-16 Solution: a4-16 =(a2)2-42 =(a2-4)(a2+4) =(a2-22)(a2+4) =(a-2)(a+2)(a2+4) a2 b2 =(ab)(a+b) Rule 3. We have to use the suitable formula.
  • 14. Q 9.Factorise: (x2+y2)2-x2y2 Solution: (x2+y2)2-x2y2 =(x2+y2)2-(xy)2 ={(x2+y2)-(xy)}{(x2+y2)+(xy)} =(x2+y2-xy)(x2+y2+xy) =(x2-xy+y2)(x2+xy+y2) a2 b2 =(ab)(a+b) Rule 3. We have to use the suitable formula.
  • 15. Q 10.Factorise: 4-(a-b)2 Solution: 4-(a-b)2 =22-(a-b)2 ={2-(a-b)}{2+(a-b)} =(2-a+b)(2+a-b) a2 b2 =(ab)(a+b) Rule 3. We have to use the suitable formula.
  • 16. Q 11. Simplify by factorization process: 722-622. Solution: 722-622 =(72-62)(72+62) =10134 =1340 We have to use the suitable formula.
  • 17. Q 12. Simplify by factorization process: 10199 Solution: 10199 = (100+1)(100-1) = (10000-1) = 9999 We have to use the suitable formula.
  • 18. Q 13. If a+b=8 and ab =15, find the value of a and b. numbers(a & b) sum(a+b) = 8 product(a.b) =15 3+5=8 1.7=7 Numbers are 3 &5. When a=3 then b=5 and when a=5 then b=3. 1 and 7 2 and 6 3 and 5 1+7=8 2+6=8 2.6=12 3.5=15 Solution: We use hit and trial method.
  • 19. Q 14. If a+b=-10 and ab=24,find the value of a and b. numbers(a & b) a.b=24 a+b=-10 -3.-8=24 -1-24=-25 Numbers are -4 & -6. When a=-4 then b=-6 and when a=-6 then b=-4. -1 and -24 -2 and -12 -3 and -8 -1.-24=24 -2.-12=24 -2-12=-14 -3-8=-11 Solution: -4 and -6 -4.-6=24 -4-6=-10 We use hit and trial method.
  • 20. Q 15. Factorize: x2+5x+6 x2 1x 1x 1x 1x 1x 1 1 1 1 1 1 x+3 x+2 x2+5x+6=(x+3)(x+2) What did we do here?
  • 21. Q 15. Factorize: x2+5x+6 Solution: x2+5x+6 Rule 4. Method 1. Multiply coefficient of x2 and constant term. 16=6(product) 2. Find the possible factors of the product 6. 6=16 6=23 3. Remember the sign of constant term. = x2+(3+2)x+6 = x2+3x+2x+6 = x(x+3)+2(x+3) = (x+3)(x+2) 4. If sign of constant term is + then choose the pair of factors whose sum is 5 (coefficient of x)
  • 22. Q 16. Resolve into factors: x2-9x+20 Rule 4. Method 1. Multiply coefficient of x2 and constant term. 120=20(product) 2. Find the possible factors of the product 20. 20=120 20=210 4. If sign of constant term is + then choose the pair of factors whose sum is 9 (coefficient of x) 20=45 Solution: x2-9x+20 = x2-(5+4)x+20 = x2-5x-4x+20 = x(x-5)-4(x-5) = (x-5)(x-4) 3. Remember the sign of constant term.
  • 23. Q 17. Factorize: x2+3x-18 Rule 4. Method 1. Multiply coefficient of x2 and constant term. 118=18(product) 2. Find the possible factors of the product 18. 18=118 18=29 18=36 Solution: x2+3x-18 =x2+(6-3)x-18 =x2+6x-3x-18 =x(x+6)-3(x+6) =(x+6)(x-3) 4. If sign of constant term is - then choose the pair of factors whose difference is 3 (coefficient of x) 3. Remember the sign of constant term.
  • 24. Q 18. Factorize: x2-5x-14 Rule 4. Method 1. Multiply coefficient of x2 and constant term. 114=14(product) 2. Find the possible factors of the product 14. 14=114 14=27 4. If sign of constant term is - then choose the pair of factors whose difference is 3 (coefficient of x) 3. Remember the sign of constant term. Solution: x2-5x-14 =x2-(7-2)x-14 =x2-7x+2x-14 =x(x-7)+2(x-7) =(x-7)(x+2)
  • 25. Q 19. Factorize: 2x2+7x+3 Rule 4. Method 1. Multiply coefficient of x2 and constant term. 23=6(product) 2. Find the possible factors of the product 6. 6=16 6=23 4. If sign of constant term is + then choose the pair of factors whose sum is 7 (coefficient of x) 3. Remember the sign of constant term. Solution: 2x2+7x+3 =2x2+7x+3 =2x2+(6+1)x+3 =2x2+6x+1x+3 =2x(x+3)+1(x+3) =(x+3)(2x+1)
  • 26. Q 20. Factorize: x2+x-30 Solution: x2+x-30 Rule 4. Method 1. Multiply coefficient of x2 and constant term. 130=30(product) 2. Find the possible factors of the product 30. 30=130 30=215 4. If sign of constant term is - then choose the pair of factors whose difference is 1 (coefficient of x). 3. Remember the sign of constant term. 30=310 30=56 =x2+x-30 =x2+(6-5)x-30 =x2+6x-5x-30 =x(x+6)-5(x+6) =(x+6)(x-5)
  • 27. Q 21. Factorize: x2+4x+4 Solution: x2+4x+4 =x2+2.x.2+22 =(x+2)2 Rule 3. We have to use the suitable formula. Rule 4. Or,
  • 28. Q 22. Factorize: 4x2-12x+9 Solution: 4x2-12x+9 = (2x)2-2.2x.3+32 = (2x-3)2 Rule 3. We have to use the suitable formula. Rule 4. Or,
  • 29. Q 23. Factorize: 4x 8 Q 24. Factorize: 4x3 - 6x2 + 8x Q 25. Factorize: xm + ym + xa + ya Q 26. Factorize: a( a + 4 ) + 2 ( a + 4 ) Q 27. Factorize: a( a 3 ) - 4( 3 a ) Q 28. Factorize: a b + 1 + ab Q 29. Factorize: x2y + xy2z + zx + yz2
  • 30. Q 30. Factorize: x2 1 Q 31. Factorize: 4x2 9b2 Q 32. Factorize: 1-36x2 Q 33. Factorize: x2-36 Q 34. Factorize: 16a2 25b2 Q 35. Factorize: 72x3 50x Q 36. Find the area of shaded part. y y 2 2
  • 31. Q 37. Factorize: x2 + 10x +25 Q 38. Factorize: p2 24p+144 Q 39. Factorize: 9a2 66a + 121 Q 40. Factorize: x2 + 9x +18 Q 41. Factorize: x2 - 9x +18 Q 42. Factorize: x2 + 3x 18 Q 43. Factorize: x2 - 3x 18 Q 44. Factorize: 15x2 x 2 Q 45. Factorize: 125x3 + 8a3 Q 46. Factorize: (x y )3 + 8(x + y)3