Class 7 Chapter 19 উৎপাদকে বিশ্লেষণ

সপ্তম শ্রেণী – অধ্যায় ১৯ : উৎপাদকে বিশ্লেষণ সম্পূর্ণ সমাধান

কষে দেখি – 19.1

 

1. নীচের বীজগাণিতিক সংখ্যামালাগুলির উৎপাদকে বিশ্লেষণ করি।

(i) 7xy    (ii) 9x2y    (iii) 16ab2c    (iv) –25lmn    (v) 12x(2 + x)    (vi) –5pq(p2 + 8)    (vii) 21xy2(3x – 2)    (viii) 121mn(m2 – n)

উত্তর –

(i) 7xy = 7 × x × y

(ii) 9x2y = 3 × 3 × x × x × y

(iii) 16ab2c = 2 × 2 × 2 × 2 × a × b × b × c

(iv) –25lmn = – 5 × 5 × l × m × n

(v) 12x(2 + x) = 3 × 2 × 2 × x × (2 + x)

(vi) –5pq(p2 + 8) = –5 × p × q × (p2 + 8)

(vii) 21xy2(3x – 2) = 3 × 7 × x × y × y × (3x – 2)

(viii) 121mn(m2 – n) = 11 × 11 × m × n × (m2 – n)

 

2. নীচের বীজগাণিতিক সংখ্যামালাগুলির সাধারণ উৎপাদক খুঁজি ও লিখি।

(i) 22xy, 33xz    

উত্তর –

22xy = 2 × 11 × x × y

33xz = 3 × 11 × x × z

22xy, 33xz –এর সাধারণ উৎপাদক = 11 × x

∴ নির্নেয় উৎপাদকগুলি হল – 11, x, 11x

(ii) 14ab2, 21ab   

উত্তর –

14ab2 = 2 × 7 × a × b × b

21ab = 3 × 7 × a × b

14ab2, 21ab –এর সাধারণ উৎপাদক = 7 × a × b

∴ নির্নেয় উৎপাদকগুলি হল – 7, a, b, 7a, 7b, ab, 7ab

(iii) –16mnl, –39nl2   

উত্তর –

–16mnl = –1 × 2 × 2 × 2 × 2 × m × n × l

–39nl2 = –1 × 3 × 13 × n × l × l

–16mnl, –39nl2 –এর সাধারণ উৎপাদক = –1 × n × l

∴ নির্নেয় উৎপাদকগুলি হল – –1, n , l, –n, –l, nl, –nl

(iv) 12a2b, 18ab2, 24abc  

উত্তর –

12a2b = 2 × 2 × 3 × a × a × b

18ab2 = 2 × 3 × 3 × a × b × b

24abc = 2 × 2 × 2 × 3 × a × b × c

12a2b, 18ab2, 24abc –এর সাধারণ উৎপাদক = 2 × 3 × a × b

∴ নির্নেয় উৎপাদকগুলি হল – 2, 3, a, b, 2a, 2b, 3a, 3b, 6a, 6b, ab, 2ab, 3ab, 6ab

(v) 2xy, 4yz, 6xz 

উত্তর –

2xy = 2 × x × y

4yz = 2 × 2 × y × z

6xz = 2 × 3 × x × z

2xy, 4yz, 6xz –এর সাধারণ উৎপাদক = 2

(vi) 18x2, 27x3, –45x    

উত্তর –

18x2 = 2 × 3 × 3 × x × x

27x3 = 3 × 3 × 3 × x × x × x

–45x = –1 × 3 × 3 × 5 × x

18x2, 27x3, –45x –এর সাধারণ উৎপাদক = 3 × 3 × x

∴ নির্নেয় উৎপাদকগুলি হল – 3, 9, x, 3x, 9x

(vii) 5mn, 6n2l2, 7l2m2

উত্তর –

5mn = 5 × m × n

6n2l2 = 2 × 3 × n × n × l × l

7l2m2 = 7 × l × l × m × m

5mn, 6n2l2, 7l2m2 –এর সাধারণ উৎপাদক = 1

 

3. দুটি বীজগাণিতিক সংখ্যামালা লিখি যাদের সাধারণ উৎপাদক –

(i) x2    (ii) 2xy    (iii) 4a2    (iv) (mn + 2)    (v) x(y + 2)

উত্তর –

(i) x2 যাদের সাধারণ উৎপাদক = x3, x4

(ii) 2xy যাদের সাধারণ উৎপাদক = 4xy, 2x2y

(iii) 4a2 যাদের সাধারণ উৎপাদক =  4a3, 8a2

(iv) (mn + 2) যাদের সাধারণ উৎপাদক =  2(mn + 2), 3m(mn + 2)  

(v) x(y + 2) যাদের সাধারণ উৎপাদক = 2x(y + 2), 3xy(y + 2)

 

4. উৎপাদকে বিশ্লেষণ করি।

(i) 5 + 10x   

উত্তর –

∴ 5 + 10x

= 5 + 2 × 5 × x

= 5(1 + 2 × x)

= 5(1 + 2x)

(ii) 2x – 6    

উত্তর –

∴ 2x – 6    

= 2 × x – 2 × 3

= 2(x – 3)

(iii) 7m – 14n   

উত্তর –

∴ 7m – 14n   

= 7 × m – 2 × 7 × n

= 7(m – 2 × n)

= 7(m – 2n)

(iv) 18xy + 21xz   

উত্তর –

∴ 18xy + 21xz   

= 2 × 3 × 3 × x × y + 3 × 7 × x × z

= 3 × x(2 × 3 × y + 7 × z)

= 3x(6y + 7z)

(v) 4xy + 6yz   

উত্তর –

∴ 4xy + 6yz   

= 2 × 2 × x × y + 2 × 3 × y × z

= 2 × y(2 × x + 3 × z)

= 2y(2x + 3z)

(vi) 7xyz – 6xy   

উত্তর –

∴ 7xyz – 6xy

= 7 × x × y × z – 2 × 3 × x × y

= x × y(7 × z – 2 × 3)

= xy(7z – 6)

(vii) 7a2 + 14a 

উত্তর –

∴ 7a2 + 14a   

= 7 × a × a + 2 × 7 × a

= 7 × a( a + 2)

= 7a(a + 2)

(viii) –15m + 20 

উত্তর –

∴ –15m + 20 

= –1 × 3 × 5 × m + 2 × 2 × 5

= 5(–1 × 3 × m + 2 × 2)

= 5(–3m + 4)

(ix) 6a2b + 8ab2   

উত্তর –

∴ 6a2b + 8ab2   

= 2 × 3 × a × a × b + 2 × 2 × 2 × a × b × b

= 2 × a × b(3 × a + 2 × 2 × b)

= 2ab(3a + 4b)

(x) 3a2 – ab2   

উত্তর –

∴ 3a2 – ab2   

= 3 × a × a – a × b × b

= a(3 × a – b × b)

= a(3a – b2)

(xi) abc – bcd

উত্তর –

∴ abc – bcd

= a × b × c – b × c × d

= b × c(a – d)

= bc(a – d)

(xii) 60xy3 + 4xy – 8   

উত্তর –

∴ 60xy3 + 4xy – 8   

= 2 × 2 × 3 × 5 × x × y × y × y + 2 × 2 × x × y – 2 × 2 × 2

= 2 × 2(3 × 5 × x × y × y × y + x × y – 2)

= 4(15 xy3 + xy – 2)

(xiii) x2yz + xy2z + xyz2   

উত্তর –

∴ x2yz + xy2z + xyz2    

= x × x × y × z + x × y × y × z + x × y × z × z

= x × y × z(x + y + z)

= xyz(x+ y +z)

(xiv) a3 – a2 + a  

উত্তর –

∴ a3 – a2 + a  

= a × a × a – a × a + a

= a (a × a – a + 1)

= a( a2 – a + 1)

(xv) x2y2z2 + x2y2 + x2y2q2

উত্তর –

∴ x2y2z2 + x2y2 + x2y2q2

= x × x × y × y × z × z + x × x × y × y + x × x × y × y × q × q

= x × x × y × y(z × z + 1 + q × q)

= x2y2(z2 + 1 + q2)

 

5. নীচের বীজগাণিতিক সংখ্যামালাগুলির উৎপাদকে বিশ্লেষণ করি।

(i) xy + 2x + y + 2   

উত্তর –

∴ xy + 2x + y + 2   

= x(y + 2) + 1(y + 2)

= (y + 2)(x + 1)

(ii) ab – 5b + a – 5   

উত্তর –

∴ ab – 5b + a – 5   

= b(a – 5) + 1(a – 5)

= (a – 5)(b + 1)

(iii) 6xy – 9y + 4x – 6    

উত্তর –

∴ 6xy – 9y + 4x – 6   

= 3y(2x – 3) + 2(2x – 3)

= (2x – 3)(3y + 2)

(iv) 15m + 9 – 35mn – 21n   

উত্তর –

∴ 15m + 9 – 35mn – 21n

= 15m – 35 mn + 9 – 21n

= 5m(3 – 7n) + 3(3 – 7n) = (3 – 7n)(5m + 3)

(v) ax + bx – ay – by 

উত্তর –

∴ ax + bx – ay – by   

= x(a + b) – y(a + b)

= (a + b)(x – y)

(vi) c – 9 + 9ab –abc 

উত্তর –

∴ c – 9 + 9ab –abc 

= c – 9 – abc + 9ab

= 1(c – 9) – ab(c – 9)

= (c – 9)(1 – ab)

কষে দেখি – 19.2

 

1. উৎপাদকে বিশ্লেষণ করি –

(i) x2 + 14x + 49                                         

উত্তর –

x2 + 14x + 49

= (x)2 + 2 × x × 7 + (7)2

= (x + 7)2

= (x + 7)(x + 7)

(ii) 4m2 – 36m + 81                                   

উত্তর –

4m2 – 36m + 81

= (2m)2 – 2 × 2m × 9 + (9)2

= (2m – 9)2

= (2m – 9)(2m – 9)

(iii) 25x2 + 30x + 9

উত্তর –

25x2 + 30x + 9

= (5x)2 + 2 × 5x × 3 + (3)2

= (5x + 3)2

= (5x + 3) (5x + 3)

(iv) 121b2 – 88bc + 16                              

উত্তর –

121b2 – 88bc + 16        

= (11b)2 – 2 × 11b × 4 + (4)2

= (11b – 4)2

= (11b – 4) (11b – 4)

(v) (x2y)2 – 4x2y2                                       

উত্তর –

(x2y)2 – 4x2y2

= x4y2– 4x2y2

= x2y2(x2 – 4)

= x × x × y × y{(x)2 – (2)2}

= xxyy(x – 2)(x + 2)

(vi) a4 + 4a2b2 + 4b4

উত্তর –

a4 + 4a2b2 + 4b4

= (a2)2 + 2 × a2 × 2b2 + (2b2)2

= (a2 + 2b2)2

= (a2 + 2b2) (a2 + 2b2)

(vii) 4x2 – 16                                               

উত্তর –

4x2 – 16

= 4(x2 – 4)

= 4{(x)2 – (2)2}

= 2 × 2(x – 2)(x + 2)

(viii) 121 – 36x2                                         

উত্তর –

121 – 36x2

= (11)2 – (6x)2

= (11 – 6x)(11 + 6x)

(ix) x2y2 – p2q2

উত্তর –

x2y2 – p2q2

= (xy)2 – (pq)2

= (xy – pq)(xy + pq)

(x) 80m2 – 125                                           

উত্তর –

80m2 – 125

= 5(16m2 – 25)

= 5{(4m)2 – (5)2}

= 5(4m – 1)(4m + 1)

(xi) ax2 – ay2                                               

উত্তর –

ax2 – ay2            

= a(x2 – y2)

= a(x – y)(x + y)

(xii) 1 – (m + n)2

উত্তর –

1 – (m + n)2

= (1)2 – (m + n)2

= {1 – (m + n)}{1 + (m + n)} = (1 – m – n)(1 + m + n)

(xiii) (2a – b – c)2 – (a – 2b – c)2            

উত্তর –

(2a – b – c)2 – (a – 2b – c)2

= {(2a – b – c) – (a – 2b – c)}{ (2a – b – c) + (a – 2b – c)}

= (2a – b – c – a + 2b + c)(2a – b – c + a – 2b – c)

= (a + b)(3a – 3b – 2c)

(xiv) x2 – 2xy – 3y2                                    

উত্তর –

x2 – 2xy – 3y2

= x2 – 2xy + y2 – 4y2

= (x – y)2 – (2y)2

= {x – y – 2y}{x – y + 2y} = (x – 3y)(x + y)

(xv) x2 + 9y2 + 6xy – z2

উত্তর –

x2 + 9y2 + 6xy – z2

= x2 + 6xy + 9y2 – z2

= (x)2 + 2 × x × 3y + (3y) – z2

= (x + 3y)2 – (z)2

= (x + 3y – z)(x + 3y + z)

(xvi) a2 – b2 + 2bc – c2                              

উত্তর –

a2 – b2 + 2bc – c2           

= a2 – (b2 – 2bc + c2)

= a2 – (b – c)2

= {a – (b – c)}{a + (b – c)}

= (a – b + c)(a + b – c)

(xvii) a2(b – c)2 – b2(c – a)2                     

উত্তর –

a2(b – c)2 – b2(c – a)2    

= {a(b – c)}2 – {b(c – a)}2

= {a(b – c) – b(c – a)} {a(b – c) + b(c – a)}

= (ab – ac – bc + ab)(ab – ac + bc – ab)

= (2ab – ac – bc)(bc – ac)

= c(b – a) (2ab – ac – bc)

(xviii) x2 –y2 –6yz –9z2

উত্তর –

x2 –y2 –6yz –9z2

= x2 – (y2 + 6yz + 9z2)

= x2 – {(y)2 + 2(y)(3z) + (3z)2}

= x2 – (y + 3z)2

= {x – (y + 3z)} {x + (y + 3z)}

= (x – y – 3z)(x + y + 3z)

(xix) x2 – y2 + 4x – 4y                                

উত্তর –

x2 – y2 + 4x – 4y             

= (x – y)(x + y) + 4(x – y)

= (x – y)(x + y + 4)

(xx) a2 – b2 + c2 – d2 – 2(ac – bd)

উত্তর –

a2 – b2 + c2 – d2 – 2(ac – bd)

= a2 – b2 + c2 – d2 – 2ac + 2bd

= a2 – 2ac + c2 – b2 + 2bd – d2

= (a2 – 2ac + c2) – (b2 – 2bd + d2)

= (a – c)2 – (b – d)2

= {(a – c) – (b – d)} {(a – c) + (b – d)}

= (a – c – b + d)(a – c + b – d)

= (a – b – c + d)(a + b – c – d)

(xxi) 2ab – a2 – b2 + c2                              

উত্তর –

2ab – a2 – b2 + c2           

= c2 – a2 + 2ab – b2

= c2 – (a2 – 2ab + b2)

= c2 – (a – b)2

= {c – (a – b)} {c + (a – b)}

= (c – a + b)(c + a – b)

= (a – b + c)( – a + b + c)

(xxii) 36x2 – 16a2 – 24ab – 9b2

উত্তর –

36x2 – 16a2 – 24ab – 9b2

= 36x2 – (16a2 – 24ab + 9b2)

= 36x2 – {(4a)2 – 2(4a)(3b) + (3b)2}

= (6x)2 – (4a – 3b)2

= {6x – (4a + 3b)} {6x + (4a + 3b)}

= (6x – 4a – 3b)(6x + 4a + 3b)

(xxiii) a2 – 1 + 2b – b2                               

উত্তর –

a2 – 1 + 2b – b2

= a2 – (1 – 2b + b2)

= a2 – (1 – b)2

= {a – (1 – b)}{a + (1 – b)}         

= (a – 1 + b)(a + 1 – b)

= (a + b – 1)(a – b + 1)

(xxiv) a2 – 2a – b2 + 2b

উত্তর –

a2 – 2a – b2 + 2b

= a2 – b2 – 2a + 2b

= (a – b)(a + b) – 2(a – b)

= (a – b)(a + b – 2)

(xxv) (a2 – b2)(c2 – d2) – 4abcd              

উত্তর –

(a2 – b2)(c2 – d2) – 4abcd           

= a2c2 – a2d2 – b2c2 + b2d2 – 4abcd

= a2c2 – 2abcd + b2d2 – a2d2 – 2abcd – b2c2

= (a2c2 – 2abcd + b2d2) – (a2d2 + 2abcd + b2c2)

= (ac – bd)2 – (ad + bc)2

= {(ac – bd) – (ad + bc)}{(ac – bd) + (ad + bc)}

= (ac – bd – ad – bc)(ac – bd + ad + bc)

(xxvi) a2 – b2 – 4ac + 4bc

উত্তর –

a2 – b2 – 4ac + 4bc

= (a – b)(a + b) – 4c(a – b)

= (a – b)(a + b – 4c)

(xxvii) (a2 – b2 – c2 + d2) – 4(ad – bc)2  

উত্তর –

(a2 – b2 – c2 + d2)2 – 4(ad – bc)2             

= (a2 – b2 – c2 + d2)2 – {2(ad – bc)}2

= {a2 – b2 – c2 + d2 – 2(ad – bc)} { a2 – b2 – c2 + d2 + 2(ad – bc)}

= {a2 – b2 – c2 + d2 – 2ad + 2bc}{{a2 – b2 – c2 + d2 + 2ad – 2bc}

= {(a2 – 2ad + d2) – (b2 – 2bc + c2)}(a2 + 2ad + d2) – (b2 + 2bc + c2)}

= {(a – d)2 – (b – c)2}{(a + d)2 – (b + c)2}

= {(a – d) – (b – c)} {(a – d) + (b – c)}{(a + d) – (b + c)} {(a + d) + (b + c)}

= (a – d – b + c)(a – d + b – c)(a + d – b – c)(a + d + b + c)

= (a – b + c – d)(a + b – c – d)(a – b – c + d)(a + b + c + d)

(xxviii) 3x2 – y2 + z2 – 2xy – 4xz

উত্তর –

3x2 – y2 + z2 – 2xy – 4xz

= 4x2 – x2 – y2 + z2 – 2xy – 4xz

= 4x2 – 4xz + z2 – (x2 + 2xy + y2)

= {(2x)2 – 2(2x)(z) + (z)2} – (x + y)2

= (2x – z)2 – (x + y)2

= {(2x – z) – (x + y)}{(2x – z) + (x + y)}

= (2x – z – x – y)(2x – z + x + y)

= (x – y – z)(3x + y – z)

 

2. উৎপাদকে বিশ্লেষণ করি –

(i) 81x4 + 4y4                                              

উত্তর –

81x4 + 4y4         

= (9x2)2 + (2y2)2

= (9x2 + 2y2)2 – 2 × 9x2 × 2y2

= (9x2 + 2y2)2 – 36x2y2

= (9x2 + 2y2)2 – (6xy)2 = (9x2 + 2y2 – 6xy)(9x2 + 2y2 + 6xy)

(ii) p4 – 13p2q2 + 4q4                                

উত্তর –

p4 – 13p2q2 + 4q4          

= p4 + 4q4 – 13p2q2       

= (p2)2 + (2q2)2 – 13p2q2

= (p2 – q2)2 + 2 × p2 × 2q2 – 13p2q2

= (p2 – q2)2 + 4p2q2 – 13p2q2

= (p2 – q2)2 – 9p2q2

= (p2 – q2)2 – (3pq)2

= (p2 – q2 + 3pq)(p2 – q2 – 3pq)

(iii) x8 – 16y8

উত্তর –

x8 – 16y8

= (x4)2 – (4y4)2

= (x4 – 4y4)(x4 + 4y4)

= {(x2)2 – (2y2)2}(x4 + 4y4)

= (x2 – 2y2)(x2 + 2y2)(x4 + 4y4)

(iv) x4 + x2y2 + y4                                       

উত্তর –

x4 + x2y2 + y4

= (x2)2 + (y2)2 + x2y2

= (x2 + y2)2 – 2x2y2 + x2y2

= (x2 + y2)2 – x2y2

= (x2 + y2)2 – (xy)2

= (x2 + y2 – xy)(x2 + y2 + xy)

(v) 3x4 + 2x2y2 – y4                                    

উত্তর –

3x4 + 2x2y2 – y4              

= 4x4 – x4 + 2x2y2 – y4

= (2x2)2 – (x4 – 2x2y2 + y4)

= (2x2)2 – {(x2)2 – 2x2y2 + (y2)2}

= (2x2)2 – (x2 – y2)2

= {2x2 – (x2 – y2)} {2x2 + (x2 – y2)}

= (2x2 – x2 + y2)(2x2 + x2 – y2)

= (x2 + y2)(3x2 – y2)

(vi) x4 + x2 + 1

উত্তর –

x4 + x2 + 1

= (x2)2 + (1)2 + x2

= (x2 + 1)2 – 2x2 + x2

= (x2 + 1)2 – x2

= (x2 + 1 – x)(x2 + 1 + x)

= (x2 – x + 1)(x2 + x + 1)

(vii) x4 + 6x2y2 + 8y4                                 

উত্তর –

x4 + 6x2y2 + 8y4

= x4 + 4x2y2 + 2x2y2 + 8y4

= x2 (x2 + 4y2) + 2y2 (x2 + 4y2)

= (x2 + 4y2) (x2 + 2y2)

(viii) 3x2 – y2 + z2 – 2xy – 4xz

উত্তর –

3x2 – y2 + z2 – 2xy – 4xz

= 4x2 – x2 – y2 + z2 – 2xy – 4xz

= (4x2 – 4xz + z2) – (x2 + 2xy + y2)

= {(2x)2 – 2(2x)(z) + (z)2} – (x + y)2

= (2x – z)2 – (x + y)2

= {(2x – z) – (x + y)}{(2x – z) + (x + y)}

= (2x – z – x – y)(2x – z + x + y)

= (x – y – z)(3x + y – z)

(ix) 3x4 – 4x2y2 + y4                                   

উত্তর –

3x4 – 4x2y2 + y4

= 4x4 – 4x2y2 + y4 – x4

= (2x2)2 – 2(2x2)(y2) + (y2)2 – x4

= (2x2 – y2)2 – (x2)2

= (2x2 – y2 – x2)(2x2 – y2 + x2)

= (x2 – y2)(3x2 – y2)

= (x – y)(x + y)(3x2 – y2)

(x) p4 – 2p2q2 – 15q4                                 

উত্তর –

p4 – 2p2q2 – 15q4           

= p4 – 2p2q2 + q2 – 16q4             

= (p2)2 – 2(p2)(q2) + (q)2 – 16q4

= (p2 – q2)2 – (4q2)2

= (p2 – q2 – 4q2)(p2 – q2 + 4q2)

= (p2 – 5q2)(p2 + 3q2)

(xi) x8 + x4y4 + y8

উত্তর –

x8 + x4y4 + y8

= (x4)2 + (y4)2 + x4y4

= (x4 + y4)2 – 2x4y4 + x4y4

= (x4 + y4)2 – x4y4

= (x4 + y4)2 – (x2y2)2

= (x4 + y4 – x2y2)(x4 + y4 + x2y2)

= (x4 + y4 – x2y2){(x2)2 + (y2)2 + x2y2}

= (x4 + y4 – x2y2){(x2 + y2)2 – 2x2y2 + x2y2}

= (x4 + y4 – x2y2){(x2 + y2)2 – x2y2}

= (x4 + y4 – x2y2){(x2 + y2)2 – (xy)2}

= (x4 + y4 – x2y2)(x2 + y2 – xy)(x2 + y2 + xy)

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