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)
;
