WBBSE Class 7 Math Solution || Ganit Prabha VII solution || কষে দেখি 19.2 সমাধান || উৎপাদকে বিশ্লেষণ (Factorization) || WBBSE Class 7 Chapter 19 Solution
Table of Contents/সুচিপত্র
ToggleWBBSE Class 7 Math Solution
Ganit Prabha VII solution
কষে দেখি 19.2 সমাধান
উৎপাদকে বিশ্লেষণ (Factorization)
WBBSE Class 7 Chapter 19 Solution
1) উৎপাদকে বিশ্লেষণ করি —
(i) x2 + 14x + 49
সমাধান:
x2 + 14x + 49
= (x)2 + 2 × x × 7 + (7)2
= (x + 7)2 বা (x + 7)(x + 7)
(ii) 4m2 – 36m + 81
সমাধান:
4m2 – 36m + 81
= (2m)2 – 2 × 2m × 9 + (9)2
= (2m – 9)2 বা (2m – 9)(2m – 9)
(iii) 25x2 + 30x + 9
সমাধান:
25x2 + 30x + 9
= (5x)2 + 2 × 5x × 3 + (3)2
= (5x + 3)2 বা (5x + 3)(5x + 3)
(iv) 121b2 – 88bc + 16
সমাধান:
121b2 – 88bc + 16
= (11b)2 – 2 × 11b × 4 + (4)2 [এখানে c নেই, সম্ভবত প্রশ্নে ভুল আছে, 16c2 হবে]
যদি প্রশ্ন 121b2 – 88bc + 16c2 হয়:
= (11b)2 – 2 × 11b × 4c + (4c)2
= (11b – 4c)2
(v) (x2y)2 – 4x2y2
সমাধান:
(x2y)2 – 4x2y2
= (x2y)2 – (2xy)2
= (x2y + 2xy)(x2y – 2xy)
= xy(x + 2) × xy(x – 2)
= x2y2(x + 2)(x – 2)
(vi) a4 + 4a2b2 + 4b4
সমাধান:
a4 + 4a2b2 + 4b4
= (a2)2 + 2 × a2 × 2b2 + (2b2)2
= (a2 + 2b2)2
(vii) 4x2 – 16
সমাধান:
4x2 – 16
= 4(x2 – 4)
= 4(x2 – 22)
= 4(x + 2)(x – 2)
(viii) 121 – 36x2
সমাধান:
121 – 36x2
= (11)2 – (6x)2
= (11 + 6x)(11 – 6x)
(ix) x2y2 – p2q2
সমাধান:
x2y2 – p2q2
= (xy)2 – (pq)2
= (xy + pq)(xy – pq)
(x) 80m2 – 125
সমাধান:
80m2 – 125
= 5(16m2 – 25)
= 5 { (4m)2 – (5)2 }
= 5(4m + 5)(4m – 5)
(xi) ax2 – ay2
সমাধান:
ax2 – ay2
= a(x2 – y2)
= a(x + y)(x – y)
(xii) 1 – (m + n)2
সমাধান:
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
সমাধান:
(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)
(xiv) x2 – 2xy – 3y2
সমাধান:
x2 – 2xy – 3y2
= x2 – 2xy + y2 – 4y2
= (x – y)2 – (2y)2
= (x – y + 2y)(x – y – 2y)
= (x + y)(x – 3y)
(xv) x2 + 9y2 + 6xy – z2
সমাধান:
x2 + 9y2 + 6xy – z2
= (x2 + 2 × x × 3y + (3y)2) – z2
= (x + 3y)2 – (z)2
= (x + 3y + z)(x + 3y – z)
(xvi) a2 – b2 + 2bc – c2
সমাধান:
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
সমাধান:
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 – ac – bc)
= c(b – a)(2ab – ac – bc)
(xviii) x2 – y2 – 6yz – 9z2
সমাধান:
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)
(xix) x2 – y2 + 4x – 4y
সমাধান:
x2 – y2 + 4x – 4y
= (x + y)(x – y) + 4(x – y)
= (x – y)(x + y + 4)
(xx) a2 – b2 + c2 – d2 – 2(ac – bd)
সমাধান:
a2 – b2 + c2 – d2 – 2(ac – bd)
= a2 – b2 + c2 – d2 – 2ac + 2bd
= (a2 – 2ac + c2) – (b2 – 2bd + d2)
= (a – c)2 – (b – d)2
= { (a – c) + (b – d) } { (a – c) – (b – d) }
= (a + b – c – d)(a – b – c + d)
(xxi) 2ab – a2 – b2 + c2
সমাধান:
2ab – a2 – b2 + c2
= c2 – (a2 – 2ab + b2)
= c2 – (a – b)2
= (c + a – b)(c – a + b)
(xxii) 36x2 – 16a2 – 24ab – 9b2
সমাধান:
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)
(xxiii) a2 – 1 + 2b – b2
সমাধান:
a2 – 1 + 2b – b2
= a2 – (b2 – 2b + 1)
= a2 – (b – 1)2
= (a + b – 1)(a – b + 1)
(xxiv) a2 – 2a – b2 + 2b
সমাধান:
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
সমাধান:
(a2 – b2)(c2 – d2) – 4abcd
= a2c2 – a2d2 – b2c2 + b2d2 – 2abcd – 2abcd
= (a2c2 – 2abcd + b2d2) – (a2d2 + 2abcd + b2c2)
= (ac – bd)2 – (ad + bc)2
= { (ac – bd) + (ad + bc) } { (ac – bd) – (ad + bc) }
= (ac + ad + bc – bd)(ac – ad – bc – bd)
(xxvi) a2 – b2 – 4ac + 4bc
সমাধান:
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
সমাধান:
(a2 – b2 – c2 + d2)2 – { 2(ad – bc) }2
= { (a2 – b2 – c2 + d2) + 2(ad – bc) } { (a2 – b2 – c2 + d2) – 2(ad – bc) }
= (a2 + d2 + 2ad – b2 – c2 – 2bc) (a2 + d2 – 2ad – b2 – c2 + 2bc)
= { (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)
(xxviii) 3x2 – y2 + z2 – 2xy – 4xz
সমাধান:
3x2 – y2 + z2 – 2xy – 4xz
= 4x2 – x2 – y2 + z2 – 2xy – 4xz
= (4x2 – 4xz + z2) – (x2 + 2xy + y2)
= (2x – z)2 – (x + y)2
= { (2x – z) + (x + y) } { (2x – z) – (x + y) }
= (3x + y – z)(x – y – z)
2) উৎপাদকে বিশ্লেষণ করি —
(i) 81x4 + 4y4
সমাধান:
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
সমাধান:
p4 – 13p2q2 + 4q4
= (p2)2 – 4p2q2 + (2q2)2 – 9p2q2
= (p2 – 2q2)2 – (3pq)2
= (p2 – 2q2 + 3pq)(p2 – 2q2 – 3pq)
(iii) x8 – 16y8
সমাধান:
x8 – 16y8
= (x4)2 – (4y4)2
= (x4 + 4y4)(x4 – 4y4)
= { (x2)2 + (2y2)2 } { (x2)2 – (2y2)2 }
= { (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 + 2y2)(x2 – 2y2)
(iv) x4 + x2y2 + y4
সমাধান:
x4 + x2y2 + y4
= (x2)2 + 2x2y2 + (y2)2 – x2y2
= (x2 + y2)2 – (xy)2
= (x2 + y2 + xy)(x2 + y2 – xy)
(v) 3x4 + 2x2y2 – y4
সমাধান:
3x4 + 2x2y2 – y4
= 4x4 – x4 + 2x2y2 – y4
= 4x4 – (x4 – 2x2y2 + y4)
= (2x2)2 – (x2 – y2)2
= { 2x2 + (x2 – y2) } { 2x2 – (x2 – y2) }
= (3x2 – y2)(x2 + y2)
= (3x2 – y2)(x2 + y2)
(vi) x4 + x2 + 1
সমাধান:
x4 + x2 + 1
= (x2)2 + 2x2 + 1 – x2
= (x2 + 1)2 – (x)2
= (x2 + x + 1)(x2 – x + 1)
(vii) x4 + 6x2y2 + 8y4
সমাধান:
x4 + 6x2y2 + 8y4
= x4 + 6x2y2 + 9y4 – y4
= (x2)2 + 2 × x2 × 3y2 + (3y2)2 – (y2)2
= (x2 + 3y2)2 – (y2)2
= (x2 + 3y2 + y2)(x2 + 3y2 – y2)
= (x2 + 4y2)(x2 + 2y2)
(viii) 3x2 – y2 + z2 – 2xy – 4xz
সমাধান:
এই প্রশ্নটি আগের 1(xxviii) প্রশ্নের হুবহু এক।
3x2 – y2 + z2 – 2xy – 4xz
= 4x2 – x2 – y2 + z2 – 2xy – 4xz
= (4x2 – 4xz + z2) – (x2 + 2xy + y2)
= (2x – z)2 – (x + y)2
= (2x – z + x + y)(2x – z – x – y) = (3x + y – z)(x – y – z)
(ix) 3x4 – 4x2y2 + y4
সমাধান:
3x4 – 4x2y2 + y4
= 4x4 – x4 – 4x2y2 + y4
= 4x4 – 4x2y2 + y4 – x4
= (2x2 – y2)2 – (x2)2
= (2x2 – y2 + x2)(2x2 – y2 – x2)
= (3x2 – y2)(x2 – y2)
= (3x2 – y2)(x + y)(x – y)
(x) p4 – 2p2q2 – 15q4
সমাধান:
p4 – 2p2q2 – 15q4
= p4 – 2p2q2 + q4 – 16q4
= (p2 – q2)2 – (4q2)2
= (p2 – q2 + 4q2)(p2 – q2 – 4q2)
= (p2 + 3q2)(p2 – 5q2)
(xi) x8 + x4y4 + y8
সমাধান:
x8 + x4y4 + y8
= (x4)2 + 2x4y4 + (y4)2 – x4y4
= (x4 + y4)2 – (x2y2)2
= (x4 + y4 + x2y2) (x4 + y4 – x2y2)
প্রথম অংশটি আবার ভাঙা যায়:
x4 + x2y2 + y4
= (x2)2 + 2x2y2 + (y2)2 – x2y2
= (x2 + y2)2 – (xy)2
= (x2 + y2 + xy)(x2 + y2 – xy)
অতএব, সম্পূর্ণ উত্তর হবে:
(x2 + y2 + xy)(x2 + y2 – xy)(x4 – x2y2 + y4)