1.উৎপাদকে বিশ্লেষণ করিঃ
(i) x2 +14x+49
সমাধানঃ
x2 +14x+49
= x2 +2.x.7 +(7)2
= (x+7)2
= (x+7)(x+7)
সূত্র –
[(a+b)2=a2+2ab+b2]
(ii) 4m2 -36m +81
সমাধানঃ
4m2 -36m +81
= (2m)2 -2 . 2m. 9 +(9)2
= (2m- 9)2
= (2m-9)(2m-9)
সূত্র –
[(a+b)2=a2+2ab+b2]
(iii) 25x2 +30x+9
সমাধানঃ
25x2 +30x+9
= (5x)2 +2.5x. 3 +(3)2
= (5x+3)2
= (5x+3)(5x+3)
সূত্র –
[(a+b)2=a2+2ab+b2]
(iv) 121b2 -88b+ 162
সমাধানঃ
121b2 -88b+ 16
= (11b)2 – 2. 11b .4 +(4)2
= (11b-4)2
= (11b -4) (11b -4)
সূত্র –
[(a-b)2=a2-2ab+b2]
(v) (x2y )2 -4x2y2
সমাধানঃ
(x2y )2 -4x2y2
= (x2y)2 –(2xy)2
= (x2y +2xy) (x2y -2xy)
= xy(x+2) ✕ xy (x-2)
= xxyy(x+2)(x-2)
সূত্র –
[a2-b2=(a+b)(a-b)]
(vi) a4 + 4a2b2 +4b4
সমাধানঃ
a4 + 4a2b2 +4b4
= (a2)2 +2. a2 . 2b2 +(2b2)2
= (a2 +2b2)2
= (a2+2b2)(a2+2b2)
সূত্র –
[(a+b)2=a2+2ab+b2]
(vii) 4x2 -16
সমাধানঃ
4x2 -16
= {(2x)2 –(4)2 }
= (2x+4)(2x-4)
= 2 ✕ 2 ✕ (x+2) ✕ (x-2)
সূত্র –
[a2-b2=(a+b)(a-b)]
(viii) 121 -36x2
সমাধানঃ
121 -36x2
= (11)2 –(6x)2
= (11+6x)(11-6x)
সূত্র –
[a2-b2=(a+b)(a-b)]
(ix) x2y2 –p2q2
সমাধানঃ
x2y2 –p2q2
= {(xy )2 –(pq)2 }
= (xy+pq)(xy-pq)
সূত্র –
[a2-b2=(a+b)(a-b)]
(x) 80m2 -125
সমাধানঃ
80m2 -125
= 5(16m2 -25)
= 5{(4m)2 –(5)2}
= 5 (4m+5) (4m-5)
সূত্র –
[a2-b2=(a+b)(a-b)]
(xi) ax2 –ay2
সমাধানঃ
ax2 –ay2
= a(x2-y2)
= a(x+y)(x-y)
সূত্র –
[a2-b2=(a+b)(a-b)]
(xii) 1- (m+n)2
সমাধানঃ
1- (m+n)2
= (1+m+n) {1-(m+n)}
= (1+m+n) (1-m-n)
সূত্র –
[a2-b2=(a+b)(a-b)]
(xiii) (2a-b-c)2 – (a-2b-c)2
সমাধানঃ
(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)
= (3a -3b -2c) (a+b)
সূত্র –
[a2-b2=(a+b)(a-b)]
(xiv) x2 -2xy -3y2
সমাধানঃ
x2 -2xy -3y2
= x2 – 2xy + y2 –4y2
= (x-y)2 –(2y)2
= (x-y+2y) (x-y-2y)
= (x+y)(x-3y)
সূত্র –
[(a+b)2=a2+2ab+b2]
(xv) x2 +9y2+6xy –z2
সমাধানঃ
x2 +9y2+6xy –z2
=x2 +6xy +9y2–z2
= x2 + 2. x .3y +(3y)2 –z2
= (x+3y)2 – z2
= (x+3y +z)(x+3y-z)
সূত্র –
[(a+b)2=a2+2ab+b2]
সূত্র –
[a2-b2=(a+b)(a-b)]
(xvi) a2-b2+2bc –c2
সমাধানঃ
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
সমাধানঃ
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)
= (bc-ac)(2ab-ac –bc)
= c(b-a)(2ab-bc-ca)
(xviii) x2 –y2 – 6yz -9z2
সমাধানঃ
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
সমাধানঃ
x2 –y2 +4x-4y
= (x+y)(x-y) +4(x-y)
= (x-y)(x+y+4)
(xx) a2-b2+c2-d2 -2(ac-bd)
সমাধানঃ
a2-b2+c2-d2 -2(ac-bd)
= a2 –b2 +c2 –d2 -2ac +2bd
= a2 -2ac +c 2 –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
সমাধানঃ
2ab –a2-b2+c2
= c2 – (a2-2ab +b2)
= c2 –(a-b)2
= {c+(a-b) }{c-(a-b)}
= (c+a-b)(c-a+b)
(xxii) 36x2 -16a2 -24ab -9b2
সমাধানঃ
36x2 -16a2 -24ab -9b2
= 36x2 – (16a2 +24ab +9b2)
= (6x)2 – {(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
সমাধানঃ
a2-1 +2b –b2
= a2 – (b2 -2b+1)
=a2 –(b-1)2
= {a+(b-1)}{a-(b-1)}
= (a+b-1)(a-b+1)
(xxiv) a2 – 2a – b2 +2b
সমাধানঃ
a2 -2a+1 –b2+2b -1
= (a2 -2a+1) –(b2 -2b+1)
=(a-1)2 – ( b-1)2
= {(a-1)+(b-1) }{(a-1)-(b-1)}
= (a-1+b-1) (a-1-b+1)
= (a+b-2) (a-b)
(xxv) (a2-b2)(c2 –d2) -4abcd
সমাধানঃ
(a2-b2)(c2 –d2) -4abcd
= a2c2 –b2c2 –a2d2 +b2d2 -2abcd-2abcd
= {(ac)2 – 2ac.bd + (bd)2 }- {(bc)2 +2 bc.ad +(ad)2}
= (ac –bd)2 – (bc +ad)2
= {(ac –bd)+(bc +ad)}{(ac –bd) – (bc +ad)}
= (ac –bd +bc +ad) (ac-bd –bc –ad)
(xxvi) a2-b2-4ac +4bc
সমাধানঃ
a2-b2-4ac +4bc
= (a+b)(a-b)-4c(a-b)
= (a-b)(a+b – 4c )
(xxvii) (a2 – b2 – c2 + d2)2 – 4(ad – bc)2
সমাধানঃ
(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
সমাধানঃ
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
সমাধানঃ
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
সমাধানঃ
p4 – 13p2q2 + 4q4
= p4 + 4q4 – 13p2q2
= (p2)2 + (2q2)2 – 13p2q2
= (p2 – 2q2)2 + 2 × p2 × 2q2 – 13p2q2
= (p2 – 2q2)2 + 4p2q2 – 13p2q2
= (p2 – 2q2)2 – 9p2q2
= (p2 – 2q2)2 – (3pq)2
= (p2 – 2q2 + 3pq)(p2 – 2q2 – 3pq)
(iii) x8 – 16y8
সমাধানঃ
x8 – 16y8
= (x4)2 – (4y4)2
= (x4 – 4y4)(x4 + 4y4)
= {(x2)2 – (2y2)2}{(x2 )2+ (2y2) 2}
= (x2 – 2y2)(x2 + 2y2){(x2 + 2y2)2 -2. x2. 2y2 }
= (x2 – 2y2)(x2 + 2y2) {(x2+2y2)2 – 4x2y2}
= (x2 – 2y2)(x2 + 2y2) {(x2+2y2)2 – (2xy)2}
= (x2 – 2y2)(x2 + 2y2) (x2+2y2 +2xy)(x2+2y2-2xy)
= (x2+2xy+2y2) (x2-2xy+ 2y2)(x2+2y2)(x2 -2y2)
(iv) x4 + x2y2 + y4
সমাধানঃ
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
সমাধানঃ
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
সমাধানঃ
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
সমাধানঃ
x4 + 6x2y2 + 8y4
= x4 + 4x2y2 + 2x2y2 + 8y4
= x2 (x2 + 4y2) + 2y2 (x2 + 4y2)
= (x2 + 4y2) (x2 + 2y2)
(viii) 3x2 – y2 + z2 – 2xy – 4xz
সমাধানঃ
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)
= (z-x+y)(z-3x-y)
(ix) 3x4 – 4x2y2 + y4
সমাধানঃ
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
সমাধানঃ
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
সমাধানঃ
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)