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Koshe dekhi 19.2 Class 7

Koshe dekhi 19.2 Class 7

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 )

 

Koshe Dekhi 19.2 Class 7

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