SSC (Class 9-10) Math BD: নবম-দশম শ্রেণি সাধারণ গণিতঃ অনুশীলনী-৩.৩ উৎপাদকে বিশ্লেষণ
ssc math solutions,class 9-10 math solution bd,ssc math pdf book, download pdf ssc/nine ten,নবম-দশম শ্রেণি সাধারণ গণিতঃ অনুশীলনী-৩.৩ উৎপাদকে বিশ্লেষণ
09 February
উৎপাদকে বিশ্লেষণঃ
উৎপাদকে বিশ্লেষণ করঃ
(১-৩০)
১.
ab(x-y)-bc(x-y)
সমাধানঃ
২.
9x2+24x+16
সমাধানঃ
9x2+24x+16
=(3x)2+2.3x.4+(4)2
=(3x+4)2
=(3x)2+2.3x.4+(4)2
=(3x+4)2
৩.
a4-27a2+1
সমাধানঃ
a4-27a2+1
=(a2)2-2.a2.1+12-25a2
=(a2-1+5a)(a2-1-5a)
=(a2)2-2.a2.1+12-25a2
=(a2-1+5a)(a2-1-5a)
৪.
x4-6x2y2+y4
সমাধানঃ
x4-6x2y2+y4
=(x2)2-2.x2.y2+(y2)2-4x2y2
=(x2-y2)2-(2xy)2
=(x2-y2-2xy)(x2-y2+2xy)
=(x2)2-2.x2.y2+(y2)2-4x2y2
=(x2-y2)2-(2xy)2
=(x2-y2-2xy)(x2-y2+2xy)
৫.
(a2-b2)(x2-y2)+4abxy
সমাধানঃ
(a2-b2)(x2-y2)+4abxy
=a2x2-b2x2-a2y2+b2y2+4abxy
=(ax)2+(by)2+2abxy-(bx)2-(ay)2+2axby
=(ax)2+(by)2+2abxy-{(bx)2+(ay)2-2axby}
=(ax+by)2-(bx-ay)2
=a2x2-b2x2-a2y2+b2y2+4abxy
=(ax)2+(by)2+2abxy-(bx)2-(ay)2+2axby
=(ax)2+(by)2+2abxy-{(bx)2+(ay)2-2axby}
=(ax+by)2-(bx-ay)2
৬.
4a2-12ab+9b2-4c2
সমাধানঃ
4a2-12ab+9b2-4c2
=(2a)2-2.2a.3b+(3b)2-(2c)2
=(2a-3b)2-(2c)2
=(2a-3b-2c)(2a-3b+2c)
=(2a)2-2.2a.3b+(3b)2-(2c)2
=(2a-3b)2-(2c)2
=(2a-3b-2c)(2a-3b+2c)
৭.
a2+6a+8-y2+2y
সমাধানঃ
a2+6a+8-y2+2y
=a2+2.a.3+32-12+2.1.y-y2
=(a+3)2-(12-2.1.y+y2)
=(a+3)2-(1-y)2
=(a+3-1+y)(a+3+1-y)
=(a+y+2)(a-y+4)
=a2+2.a.3+32-12+2.1.y-y2
=(a+3)2-(12-2.1.y+y2)
=(a+3)2-(1-y)2
=(a+3-1+y)(a+3+1-y)
=(a+y+2)(a-y+4)
৮.
16x2-25y2-8xz+10yz
সমাধানঃ
16x2-25y2-8xz+10yz
=(4x)2-(5y)2-2z(4x-5y)
=(4x-5y)(4x+5y)-2z(4x-5y)
=(4x-5y)(4x+5y-2z)
=(4x)2-(5y)2-2z(4x-5y)
=(4x-5y)(4x+5y)-2z(4x-5y)
=(4x-5y)(4x+5y-2z)
৯.
2b2c2+2c2a2+2a2b2-a4-b4-c4
সমাধানঃ
2b2c2+2c2a2+2a2b2-a4-b4-c4
=4a2b2-a4-b4-c4-2a2b2+2b2c2+2c2a2
=(2ab)2-{(a2)2+(b2)2+(c2)2+2a2b2+2b2 (-c)2+2(-c)2a2}
=(2ab)2-(a2+b2-c2)2
=(2ab-a2-b2+c2)(2ab+a2+b2-c2)
={c2-(a2+b2-2ab)}{(a+b)2-c2}
={c2-(a-b)2}{a+b-c)(a+b+c)
=(c-a+b)(c+a-b)(a+b-c)(a+b+c)
=4a2b2-a4-b4-c4-2a2b2+2b2c2+2c2a2
=(2ab)2-{(a2)2+(b2)2+(c2)2+2a2b2+2b2 (-c)2+2(-c)2a2}
=(2ab)2-(a2+b2-c2)2
=(2ab-a2-b2+c2)(2ab+a2+b2-c2)
={c2-(a2+b2-2ab)}{(a+b)2-c2}
={c2-(a-b)2}{a+b-c)(a+b+c)
=(c-a+b)(c+a-b)(a+b-c)(a+b+c)
১০.
x2+13x+36
সমাধানঃ
x2+13x+36
=x2+9x+4x+36
=x(x+9)+4(x+9)
=(x+9)((x+4)
=x2+9x+4x+36
=x(x+9)+4(x+9)
=(x+9)((x+4)
১১.
x4+x2-20
সমাধানঃ
x4+x2-20
=x4+5x2-4x2-20
=x2(x2+5)-4(x2+5)
=(x2-4)(x2+5)
=(x2-22)(x2+5)
=(x-2)(x+2)(x2+5)
=x4+5x2-4x2-20
=x2(x2+5)-4(x2+5)
=(x2-4)(x2+5)
=(x2-22)(x2+5)
=(x-2)(x+2)(x2+5)
১২.
a2-30a+216
সমাধানঃ
a2-30a+216
=a2-18a-12a+216
=a(a-18)-12(a-18)
=(a-12)(a-18)
=a2-18a-12a+216
=a(a-18)-12(a-18)
=(a-12)(a-18)
১৩.
a8-a4-2
সমাধানঃ
a8-a4-2
=a8-2a4+a4-2
=a4(a4-2)+1(a4-2)
=(a4+1)(a4-2)
=a8-2a4+a4-2
=a4(a4-2)+1(a4-2)
=(a4+1)(a4-2)
১৪.
x2-37x-650
সমাধানঃ
x2-37x-650
=x2+13x-50x-650
=x(x+13)-50(x-13)
=(x-50)(x+13)
=x2+13x-50x-650
=x(x+13)-50(x-13)
=(x-50)(x+13)
১৫.
9x2y2-5xy2-14y2
সমাধানঃ
9x2y2-5xy2-14y2
=y2(9x2-5x-14)
=y2(9x2-14x+9x-14)
=y2{x(9x-14)+1(9x-14)}
=y2(x+1)(9x-14)
=y2(9x2-5x-14)
=y2(9x2-14x+9x-14)
=y2{x(9x-14)+1(9x-14)}
=y2(x+1)(9x-14)
১৬.
4x4-27x2-81
সমাধানঃ
4x4-27x2-81
=4x4-36x2+9x2-81
=4x2(x2-9)+9(x2-9)
=(4x2+9)(x2-9)
=(4x2+9)(x2-32)
=(4x2+9)(x-3)(x+3)
=4x4-36x2+9x2-81
=4x2(x2-9)+9(x2-9)
=(4x2+9)(x2-9)
=(4x2+9)(x2-32)
=(4x2+9)(x-3)(x+3)
১৭.
ax2+(a2+1)x+a
সমাধানঃ
ax2+(a2+1)x+a
=ax2+a2x+x+a
=ax(x+a)+1(x+a)
=(ax+1)(x+a)
=ax2+a2x+x+a
=ax(x+a)+1(x+a)
=(ax+1)(x+a)
১৮.
3(a2+2a)2-22(a2+2a)+40
সমাধানঃ
3(a2+2a)2-22(a2+2a)+40
মনে করি, a2+2a=x
∴প্রদত্ত রাশি
=3x2-12x-10x+40
=3x(x-4)-10(x-4)
=(3x-10)(x-4)
={3.(a2+2a)-10}{a2+2a-4} [মান বসিয়ে]
=(3a2+6a-10)(a2+2a-4)
মনে করি, a2+2a=x
∴প্রদত্ত রাশি
=3x2-12x-10x+40
=3x(x-4)-10(x-4)
=(3x-10)(x-4)
={3.(a2+2a)-10}{a2+2a-4} [মান বসিয়ে]
=(3a2+6a-10)(a2+2a-4)
১৯.
14(x+z)2-29(x+z)(x+1)-15(x+1)2
সমাধানঃ
14(x+z)2-29(x+z)(x+1)-15(x+1)2
∴প্রদত্ত রাশি
=14a2-35ab+6ab-15b2
=7a(2a-5b)+3b(2a-5b)
=(7a+3b)(2a-5b)
={7(x+z)+3(x+1)}{2(x+z)-5(x+1)} [মান বসিয়ে]
=(7x+7z+3x+3)(2x+2z-5x-5)
=(10x+7z+3)(2z-3x-5)
∴প্রদত্ত রাশি
=14a2-35ab+6ab-15b2
=7a(2a-5b)+3b(2a-5b)
=(7a+3b)(2a-5b)
={7(x+z)+3(x+1)}{2(x+z)-5(x+1)} [মান বসিয়ে]
=(7x+7z+3x+3)(2x+2z-5x-5)
=(10x+7z+3)(2z-3x-5)
২০.
(a-1)x2+a2xy+(a+1)y2
সমাধানঃ
(a-1)x2+a2xy+(a+1)y2
মনে করি,
a-1=p
এবং, a+1=q
তাহলে, (a-1)(a+1)=pq
বা, a2-1=pq
বা, a2=pq+1
∴প্রদত্ত রাশি
=px2+pqxy+xy+qy2
=px(x+qy)+y(x+qy)
=(px+y)(x+qy)
={(a-1)x+y}{x+(a+1)y} [মান বসিয়ে]
=(ax-x+y)(x+ay+y)
মনে করি,
a-1=p
এবং, a+1=q
তাহলে, (a-1)(a+1)=pq
বা, a2-1=pq
বা, a2=pq+1
∴প্রদত্ত রাশি
=px2+pqxy+xy+qy2
=px(x+qy)+y(x+qy)
=(px+y)(x+qy)
={(a-1)x+y}{x+(a+1)y} [মান বসিয়ে]
=(ax-x+y)(x+ay+y)
২১.
x3+3x2+3x+2
সমাধানঃ
x3+3x2+3x+2
=x3+3.x2.1+3.x.12+13+1
=(x+1)3+13
={(x+1)+1}{(x+1)2-(x+1).1+12}
=(x+2)(x2+2x+1-x-1+1)
=(x+2)(x2+x+1)
=x3+3.x2.1+3.x.12+13+1
=(x+1)3+13
={(x+1)+1}{(x+1)2-(x+1).1+12}
=(x+2)(x2+2x+1-x-1+1)
=(x+2)(x2+x+1)
২২.
a3-6a2+12a-9
সমাধানঃ
a3-6a2+12a-9
=a3-3.a2.2+3.a.(2)2-23-1
=(a-2)3-13
=(a-2-1){(a-2)2+(a-2).1+12}
=(a-3)(a2-2.a.2+22+a-2+1)
=(a-3)(a2-4a+8+a-2+1)
=(a-3)((a2-3a+3)
=a3-3.a2.2+3.a.(2)2-23-1
=(a-2)3-13
=(a-2-1){(a-2)2+(a-2).1+12}
=(a-3)(a2-2.a.2+22+a-2+1)
=(a-3)(a2-4a+8+a-2+1)
=(a-3)((a2-3a+3)
২৩.
a3-9b3+(a+b)3
সমাধানঃ
a3-9b3+(a+b)3
=a3-b3+(a+b)3-8b3
=(a-b)(a2+ab+b2)+(a+b)3-(2b)3
=(a-b)(a2+ab+b2)+(a+b-2b){(a+b)2+(a+b).2b+(2b)2}
=(a-b)(a2+ab+b2)+(a-b)(a2+b2+2ab+2ab+2b2+4b2)
=(a-b)(a2+ab+b2)+(a-b)(a2+4ab+7b2)
=(a-b)( a2+ab+b2+ a2+4ab+7b2)
=(a-b)(2a2+5ab+8b2)
=a3-b3+(a+b)3-8b3
=(a-b)(a2+ab+b2)+(a+b)3-(2b)3
=(a-b)(a2+ab+b2)+(a+b-2b){(a+b)2+(a+b).2b+(2b)2}
=(a-b)(a2+ab+b2)+(a-b)(a2+b2+2ab+2ab+2b2+4b2)
=(a-b)(a2+ab+b2)+(a-b)(a2+4ab+7b2)
=(a-b)( a2+ab+b2+ a2+4ab+7b2)
=(a-b)(2a2+5ab+8b2)
২৪.
8x3+12x2+6x-63
সমাধানঃ
8x3+12x2+6x-63
=(2x)3+3.(2x)2.1+3.2x.12+13+6x-64
=(2x+1)3-64
=(2x+1)3-43
=(2x+1-4){(2x+1)2+(2x+1)4+42)
=(2x+1)(4x2+4x+1+8x+4+16)
=(2x+1)(4x2+12x+21)
=(2x)3+3.(2x)2.1+3.2x.12+13+6x-64
=(2x+1)3-64
=(2x+1)3-43
=(2x+1-4){(2x+1)2+(2x+1)4+42)
=(2x+1)(4x2+4x+1+8x+4+16)
=(2x+1)(4x2+12x+21)
২৫.
8a3+b3/27
সমাধানঃ
8a3+b3/27
=(2a)3+(b/3)3
=(2a+b/3)+{(2a)2-2a.b/3+(b/3)2}
=(2a+b/3)(4a2-2ab/3+b2/9)
={(6a+b)/3}{36a2-6ab+b2)/9}
=(6a+b)(36a2-6ab+b2)/27
=(2a)3+(b/3)3
=(2a+b/3)+{(2a)2-2a.b/3+(b/3)2}
=(2a+b/3)(4a2-2ab/3+b2/9)
={(6a+b)/3}{36a2-6ab+b2)/9}
=(6a+b)(36a2-6ab+b2)/27
২৬.
a6/27-b6
সমাধানঃ
a6/27-b6
=(a2/3)3-(b2)3
=(a2/3-b2){(a2/3)2+(a2/3).b2+(b2)2}
=(a2/3-b2)(a4/9+a2b2/3+b4)
=(a2/3)3-(b2)3
=(a2/3-b2){(a2/3)2+(a2/3).b2+(b2)2}
=(a2/3-b2)(a4/9+a2b2/3+b4)
২৭.
4a2+1/4a2-2+4a-1/a
সমাধানঃ
4a2+1/4a2-2+4a-1/a
=(2a)2+(1/2a)2-2.2a.(1/2a)+2(2a-1/2a)
=(2a-1/2a)2+2(2a-1/2a)
=(2a-1/2a)2(2a-1/2a+2)
=(2a)2+(1/2a)2-2.2a.(1/2a)+2(2a-1/2a)
=(2a-1/2a)2+2(2a-1/2a)
=(2a-1/2a)2(2a-1/2a+2)
২৮.
(3a+1)3-(2a-3)3
সমাধানঃ
(3a+1)3-(2a-3)3
={(3a+1)-(2a-3)}{(3a+1)2+(3a+1)(2a-3)+(2a-3)2}
=(3a+1-2a+3){(3a)2+2.3a.1+12+3a.2a+2a-3a.3-3+(2a)2-2.2a.3+32}
=(a+4)(9a2+6a+1+6a2+2a-9a-3+4a2-12a+9)
=(a+4)(19a2-13a+7)
={(3a+1)-(2a-3)}{(3a+1)2+(3a+1)(2a-3)+(2a-3)2}
=(3a+1-2a+3){(3a)2+2.3a.1+12+3a.2a+2a-3a.3-3+(2a)2-2.2a.3+32}
=(a+4)(9a2+6a+1+6a2+2a-9a-3+4a2-12a+9)
=(a+4)(19a2-13a+7)
২৯.
(x+2)(x+3)(x+4)(x+5)-48
সমাধানঃ
(x+2)(x+3)(x+4)(x+5)-48
=(x+2)(x+5)(x+3)(x+4)-48
=(x2+2x+5x+10)(x2+3x+4x+12)-48
=(x2+7x+10)(x2+7x+12)-48
মনে করি, x2+7x+10=m
∴প্রদত্ত রাশি
=m2+2m-48
=m2+8m-69m-48
=m(m+8)-6(m+8)
=(m-6)(m+8)
=(x2+7x+10-6)(x2+7x+10+8) [মান বসিয়ে]
=(x2+7x+4)(x2+7x+18)
=(x+2)(x+5)(x+3)(x+4)-48
=(x2+2x+5x+10)(x2+3x+4x+12)-48
=(x2+7x+10)(x2+7x+12)-48
মনে করি, x2+7x+10=m
∴প্রদত্ত রাশি
=m2+2m-48
=m2+8m-69m-48
=m(m+8)-6(m+8)
=(m-6)(m+8)
=(x2+7x+10-6)(x2+7x+10+8) [মান বসিয়ে]
=(x2+7x+4)(x2+7x+18)
৩০.
(x-1)(x-3)(x-5)(x-7)-65
সমাধানঃ
(x-1)(x-3)(x-5)(x-7)-65
=(x-1)(x-7)(x-3)(x-5)-65
=(x2-7x-x+7)(x2-5x-3x+15)-65
=(x2-8x+7)(x2-8x+15)-65
ধরি, x2-8x+7=p
∴প্রদত্ত রাশি
=p2+8p-65
=p2+13p-5p-65
=p(p+13)-5(p+13)
=(p-5)(p+13)
=(x2-8x+7-5)(x2-8x+7+13) [মান বসিয়ে]
=(x2-8x+2)(x2-8x+20)
=(x-1)(x-7)(x-3)(x-5)-65
=(x2-7x-x+7)(x2-5x-3x+15)-65
=(x2-8x+7)(x2-8x+15)-65
ধরি, x2-8x+7=p
∴প্রদত্ত রাশি
=p2+8p-65
=p2+13p-5p-65
=p(p+13)-5(p+13)
=(p-5)(p+13)
=(x2-8x+7-5)(x2-8x+7+13) [মান বসিয়ে]
=(x2-8x+2)(x2-8x+20)
৩১. দেখাও যে,
(x+1)(x+2)(3x-1)(3x-4)=(3x2+2x-1)(3x2+2x-8)
সমাধানঃ
বামপক্ষ
=(x+1)(x+2)(3x-1)(3x-4)
=(x+1)(3x-1)(x+2)(3x-4)
=(3x2+3x-x-1)(3x2+6x-4x-8)
=(3x2+2x-1)(3x2+2x-8)
=ডানপক্ষ (দেখানো হলো)
=(x+1)(x+2)(3x-1)(3x-4)
=(x+1)(3x-1)(x+2)(3x-4)
=(3x2+3x-x-1)(3x2+6x-4x-8)
=(3x2+2x-1)(3x2+2x-8)
=ডানপক্ষ (দেখানো হলো)
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