Koshe dekhi 19.2 Class 7

1) উৎপাদকে বিশ্লেষণ করি –

(i) x² + 14x + 49

সমাধানঃ

x² + 14x + 49

= x² + 7x + 7x + 49

= x (x + 7) + 7 (x + 7)

= (x + 7) (x + 7)

(ii) 4m² − 36m + 81

সমাধানঃ

4m² − 36m + 81

= 4m² − 18m − 18m + 81

= 2m (2m − 9) −9 (2m − 9)

= (2m − 9) (2m − 9)

(iii) 25x² + 30x + 9

সমাধানঃ

25x² + 30x + 9

= 25x² + 15x + 15x + 9

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

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

(iv) 121b² − 88b + 16

সমাধানঃ

121b² − 88bc + 16

= 121b² − 44b − 44b + 16

= 11b (11b − 4) − 4 (11b − 4)

= (11b − 4) (11b − 4)

(v) (x²y)² −4x²y²

সমাধানঃ

(x²y)² −4x²y²

= (x²y)² − (2xy)²

= (x²y + 2xy) (x²y − 2xy)

= xy(x + 2) xy(x − 2)

= x² y² (x + 2) (x − 2)

(vi) a⁴+ 4a²b²+ 4b⁴

সমাধানঃ

a⁴+ 4a²b²+ 4b⁴

= (a²)² + 2 × a² × 2b² + (2b²)²

= ( a²+ 2b²)²

= ( a²+ 2b²) ( a²+ 2b²)

(vii) 4x² − 16

সমাধানঃ

4x² − 16

= (2x)² − (4)²

= (2x + 4) (2x − 4)

= 2 (x + 2) 2 (x − 2)

= 4(x + 2) (x − 2)

(viii) 121 − 36x²

সমাধানঃ

121 − 36x²

= (11)² − (6x)²

= (11 + 6x) (11 − 6x)

(ix) x²y² − p²q²

সমাধানঃ

x²y² − p²q²

= (xy)² − (pq)²

= (xy + pq) (xy − pq)

(x) 80m² − 125

সমাধানঃ

80m² − 125

= 5(16m² − 25)

= 5[(4m)² − (5)²]

= 5 ( 4m + 5) (4m − 5)

(xi) ax² − ay²

সমাধানঃ

ax² − ay²

= a(x² − y²)

= a(x + y) ( x − y)

(xii) 1 − (m + n)²

সমাধানঃ

1 − (m + n)²

= 1² − (m + n)²

= (1 + m + n) (1 − m − n)

(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)

= (3a − 3b − 2c) (a + b)

(xiv) x² − 2xy − 3y²

সমাধানঃ

x² − 2xy − 3y²

= x² − 3xy + xy − 3y²

= x(x − 3y) + y (x − 3y)

= (x − 3y) (x + y)

(xv) x²+ 9y²+ 6xy − z²

সমাধানঃ

x²+ 9y²+ 6xy − z²

= x²+ 2.x.3y + (3y)² − z²

= ( x + 3y)² − z²

= (x + 3y + z) (x + 3y − z)

(xvi) a²− b² + 2bc − c²

সমাধানঃ

a²− b² + 2bc − c²

= a²− ( b² − 2bc + c²)

= a²− (b − c)²

= (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

= (ab − ac)² − (bc − ab)²

= (ab − ac + bc − ab) (ab − ac − bc + ab)

= (bc − ac) (2ab − ac − bc)

= c (b − a) (2ab − ac − bc)

(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)

(xix) x² − y²+ 4x − 4y

সমাধানঃ

x² − y²+ 4x − 4y

= (x + y) (x − y) + 4(x − y)

= (x − y) [(x + y) + 4]

= (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+ c²) − (b² − 2bd + d²)

= (a − c)² − (b − d)²

= (a − c + b − d) ( a − c − b + 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)

(xxii) 36x² − 16a² − 24ab − 9b²

সমাধানঃ

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

সমাধানঃ

a²− 1 + 2b − b²

= (a)² − [1 −2b + b²]

= (a)² − (1 − b)²

= (a + 1 − b) (a − 1 + b)

(xxiv) a² − 2a − b²+ 2b

সমাধানঃ

a² − 2a − b²+ 2b

= a² − b²− 2a + 2b

$=\left ( a+b \right )\left ( a-b \right )-2\left ( a-b \right )$

$=\left ( a-b \right )\left [\left ( a+b \right )-2 \right ]$

$=\left ( a-b \right )\left ( a+b-2 \right )$

(xxv) (a² − b²) (c² − d²) − 4abcd

সমাধানঃ

(a² − b²) (c² − d²) − 4abcd

$=a^{2}c^{2}-a^{2}d^{2}-b^{2}c^{2}+b^{2}d^{2}-4abcd$

$=a^{2}c^{2}+b^{2}d^{2}-\left [a^{2}d^{2}+b^{2}c^{2} \right ]-2abcd-2abcd$

$=\left [\left ( ac \right )^{2}-2\times ac\times bd+\left ( bd \right )^{2} \right ]-\left [ \left ( ad \right )^{2}+2\times ad\times bc+\left ( bc \right )^{2} \right ]$

$=\left ( ac-bd \right )^{2}-\left ( ad+bc \right )^{2}$

$=\left ( ac-bd+ad+bc \right )\left ( ac-bd-ad-bc \right )$

(xxvi) a² − b² − 4ac + 4bc

সমাধানঃ

a² − b² − 4ac + 4bc

= (a² − b²) − 4c (a − b)

= (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 − 4(ad − bc)²

= (a² − b² − c² + d²)² − (2ad − 2bc)²

= (a² − b² − c² + d² + 2ad −2bc) (a² − b² − c² + d² −2ad + 2bc)

$=\left [ a^{2}+2ad+d^{2}-\left ( b^{2}+2bc+c^{2} \right ) \right ]\left [ a^{2}-2ad+d^{2}-\left ( b^{2}-2bc+c^{2} \right ) \right ]$

$=\left [ \left ( a+d \right )^{2}-\left ( b+c \right )^{2} \right ]\left [ \left ( a-d \right )^{2}-\left ( b-c \right )^{2} \right ]$

$=\left [ \left (a+d+b+c \right )\left ( a+d-b-c \right ) \right ]\left [ \left (a-d+b-c \right )\left ( a-d-b+c \right ) \right ]$

$=\left (a+d+b+c \right )\left ( a+d-b-c \right )\left (a-d+b-c \right )\left ( a-d-b+c \right )$

(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 − z)² − (x + y)²

= (2x − z + x + y) (2x − z − x − y)

= (3x − z + y) (x − z − y)

2) উৎপাদকে বিশ্লেষণ করি –

(i) 81x⁴ + 4y⁴

সমাধানঃ

81x⁴ + 4y⁴

$=\left ( 9x^{2} \right )^{2}+\left ( 2y^{2} \right )^{2}$

$=\left ( 9x^{2} +2y^{2}\right )^{2}-2\times 9x^{2}\times 2y^{2}$

$=\left ( 9x^{2} +2y^{2}\right )^{2}-36x^{2}y^{2}$

$=\left ( 9x^{2} +2y^{2}\right )^{2}-\left ( 6xy \right )^{2}$

$=\left ( 9x^{2} +2y^{2}+6xy\right )\left (9x^{2} +2y^{2}-6xy \right )$

(ii) p⁴ − 13p²q² + 4q⁴

সমাধানঃ

p⁴ − 13p²q² + 4q⁴

$=\left ( p^{2} \right )^{2}-2\times p^{2}\times 2q^{2}+\left ( 2q^{2} \right )^{2}-9p^{2}q^{2}$

$=\left ( p^{2}-2q^{2} \right )^{2}-9p^{2}q^{2}$

$=\left ( p^{2}-2q^{2} \right )^{2}-\left (3pq \right )^{2}$

$=\left ( p^{2}-2q^{2}+3pq \right )\left (p^{2}-2q^{2}-3pq \right )$

(iii) x⁸ − 16y⁸

সমাধানঃ

x⁸ − 16y⁸

$=\left ( x^{4} \right )^{2}-\left ( 4y^{4} \right )^{2}$

$=\left ( x^{4}+4y^{4} \right )\left ( x^{4}-4y^{4} \right )$

$=\left [\left ( x^{2} \right )^{2}+\left ( 2y^{2} \right )^{2} \right ]\left [\left ( x^{2} \right )^{2}-\left ( 2y^{2} \right )^{2} \right ]$

$=\left [\left ( x^{2}+2y^{2} \right )^{2}-2\times x^{2}\times 2y^{2} \right ]\left [\left ( x^{2}+2y^{2} \right )\left ( x^{2}-2y^{2} \right ) \right ]$

$=\left [\left ( x^{2}+2y^{2} \right )^{2}-4x^{2}y^{2} \right ]\left [\left ( x^{2}+2y^{2} \right )\left ( x^{2}-2y^{2} \right ) \right ]$

$=\left [\left ( x^{2}+2y^{2} \right )^{2}-\left ( 2xy \right )^{2} \right ]\left [\left ( x^{2}+2y^{2} \right )\left ( x^{2}-2y^{2} \right ) \right ]$

$=\left [\left ( x^{2}+2y^{2}+2xy \right )\left ( x^{2}+2y^{2}-2xy \right ) \right ]\left [\left ( x^{2}+2y^{2} \right )\left ( x^{2}-2y^{2} \right ) \right ]$

$=\left ( x^{2}+2xy+2y^{2} \right )\left ( x^{2}-2xy+2y^{2} \right )\left ( x^{2}+2y^{2} \right )\left ( x^{2}-2y^{2} \right )$

(iv) x⁴+ x²y²+ y⁴

সমাধানঃ

x⁴+ x²y²+ y⁴

$=\left ( x^{2} \right )^{2}+2\times x^{2}\times y^{2}+\left ( y^{2} \right )^{2}-x^{2}y^{2}$

$=\left ( x^{2}+y^{2} \right )^{2}-\left ( xy \right )^{2}$

$=\left ( x^{2}+y^{2}+xy \right )\left ( x^{2}+y^{2}-xy \right )$

(v) 3x⁴+ 2x²y² − y⁴

সমাধানঃ

3x⁴+ 2x²y² − y⁴

= 4x⁴− x⁴+ 2x²y² − y⁴

$=\left ( 2x^{2} \right )^{2}-\left ( x^{4}-2x^{2}y^{2}+y^{4} \right )$

$=\left ( 2x^{2} \right )^{2}-\left [ \left (x^{2} \right )^{2}-2x^{2}y^{2}+\left (y^{2} \right )^{2} \right ]$

$=\left ( 2x^{2} \right )^{2}-\left (x^{2} -y^{2} \right )^{2}$

$=\left ( 2x^{2}+x^{2} -y^{2} \right )\left (2x^{2}-x^{2} +y^{2} \right )$

$=\left ( 3x^{2}-y^{2} \right )\left ( x^{2}+y^{2} \right )$

(vi) x⁴ + x²+ 1

সমাধানঃ

x⁴ + x²+ 1

$=\left ( x^{2} \right )^{2}+2x^{2}+\left ( 1 \right )^{2}-x^{2}$

$=\left ( x^{2}+1 \right )^{2}-\left ( x \right )^{2}$

$=\left ( x^{2}+1+x \right )\left (x^{2}+1-x \right )$

(vii) x⁴+ 6x²y² + 8y⁴

সমাধানঃ

x⁴+ 6x²y² + 8y⁴

= x⁴+ 2x²y² + 4x²y²+ 8y⁴

$=x^{2}\left ( x^{2}+2y^{2} \right )+4y^{2}\left ( x^{2}+2y^{2} \right )$

$=\left ( x^{2}+4y^{2} \right )\left ( x^{2}+2y^{2} \right )$

(viii) 3x² −y² + z² − 2xy − 4xz

সমাধানঃ

3x² −y² + z² − 2xy − 4xz

= 4x² −x² −y² + z² − 2xy − 4xz

= 4x² − 4xz + z²− x² − 2xy − y²

= 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

সমাধানঃ

3x⁴ − 4x²y²+ y⁴

= 4x⁴ −x⁴ − 4x²y²+ y⁴

$=\left ( 2x^{2} \right )^{2}-2\times 2x^{2}\times y^{2} +\left ( y^{2} \right )^{2}-x^{4}$

$=\left ( 2x^{2}-y^{2} \right )^{2}-\left ( x^{2} \right )^{2}$

$=\left ( 2x^{2}-y^{2}+x^{2} \right )\left ( 2x^{2}-y^{2}-x^{2} \right )$

$=\left ( 3x^{2}-y^{2} \right )\left ( x^{2}-y^{2} \right )$

$=\left ( 3x^{2}-y^{2} \right )\left ( x+y \right )\left ( x-y \right )$

(x) p⁴ − 2p²q² −15q⁴

সমাধানঃ

p⁴ − 2p²q² −15q⁴

$=\left ( p^{2} \right )^{2}-2\times p^{2}\times q^{2}+\left ( q^{2} \right )^{2}-16q^{4}$

$=\left ( p^{2}-q^{2} \right )^{2}-\left ( 4q^{2} \right )^{2}$

$=\left ( p^{2}-q^{2}+4q^{2} \right )\left ( p^{2}-q^{2}-4q^{2} \right )$

$=\left ( p^{2}+3q^{2} \right )\left ( p^{2}-5q^{2} \right )$

(xi) x⁸+ x⁴y⁴+ y⁸

সমাধানঃ

x⁸+ x⁴y⁴+ y⁸

$=\left ( x^{4} \right )^{2}+2x^{4}y^{4}+\left ( y^{4} \right )^{2}-x^{4}y^{4}$

$=\left ( x^{4}+y^{4} \right )^{2}-\left (x^{2}y^{2} \right )^{2}$

$=\left ( x^{4}+y^{4}+x^{2}y^{2} \right )\left (x^{4}+y^{4}-x^{2}y^{2} \right )$

$=\left [ \left ( x^{2} \right )^{2}+2x^{2}y^{2}+\left ( y^{2} \right )^{2}-x^{2}y^{2} \right ]\left (x^{4}+y^{4}-x^{2}y^{2} \right )$

$=\left [ \left ( x^{2}+y^{2} \right )^{2}-\left (xy \right )^{2} \right ]\left (x^{4}+y^{4}-x^{2}y^{2} \right )$

$=\left [ \left ( x^{2}+y^{2}+xy \right )\left (x^{2}+y^{2}-xy \right ) \right ]\left (x^{4}+y^{4}-x^{2}y^{2} \right )$

$=\left ( x^{2}+y^{2}+xy \right )\left (x^{2}+y^{2}-xy \right ) \left (x^{4}+y^{4}-x^{2}y^{2} \right )$