SSC/Class 9-10 Math Solution Exercise 3.3

উৎপাদকে বিশ্লেষণ কর: (১-৩০)

১. ab(x – y) – bc(x – y)

সমাধান: ab(x – y) – bc(x – y)
= (x – y)(ab – bc)
= (x – y){b(a – c)}
= b(x – y)(a – c)

২. 9x² + 24x + 16

সমাধান: 9x² + 24x + 16
= (3x)² + 2.3x.4 + (4)²
= (3x + 4)²

৩. a⁴ – 27a² + 1

সমাধান: a⁴ – 27a² + 1
= (a²)² – 2.a².1 + 1²– 25a²
= (a² – 1)² – (5a)²
= (a² – 1 + 5a)(a²  -1 -5a)

৪. x⁴ – 6x²y² + y⁴

সমাধান: x⁴ – 6x²y² + y⁴
= (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

সমাধান: (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
= (ax + by + bx – ay)(ax + by – bx + ay)

৬. 4a²-12ab+9b²-4c²

সমাধান: 4a² – 12ab + 9b² – 4c²
= (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

সমাধান: 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)

৮. 16x² – 25y² – 8xz + 10yz

সমাধান: 16x² – 25y² – 8xz + 10yz
= (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

সমাধান: 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)

১০. x² + 13x + 36

সমাধান: x² + 13x + 36
= x² + 9x + 4x + 36
= x(x + 9) + 4(x + 9)
= (x + 9)(x+  4)

১১. x⁴ + x² – 20

সমাধান: 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)

১২. a² – 30a + 216

সমাধান: a² – 30a + 216
= a² – 18a – 12a + 216
= a(a – 18) – 12(a – 18)
= (a – 12)(a – 18)

১৩. a⁸ – a⁴ – 2

সমাধান: a⁸ – a⁴ – 2
= a⁸ – 2a⁴ + a⁴ – 2
= a⁴(a⁴ – 2) + 1(a⁴ – 2)
= (a⁴ + 1)(a⁴ – 2)

১৪. x² – 37x – 650

সমাধান: x² – 37x – 650
= x² + 13x – 50x – 650
= x(x + 13) – 50(x – 13)
= (x – 50)(x + 13)

১৫. 9x²y² – 5xy² – 14y²

সমাধান: 9x²y² – 5xy² – 14y²
= 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

সমাধান: 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)

১৭. ax² + (a² + 1)x + a

সমাধান: ax² + (a² + 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

সমাধান: 3(a² + 2a)² – 22(a² + 2a) + 40
মনে করি, a² + 2a = x
∴ প্রদত্ত রাশি = 3x² – 22x + 40
= 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

সমাধান: 14(x + z)² – 29(x + z)(x + 1) – 15(x + 1)²
মনে করি, x + z  = a; x + 1 = b
∴ প্রদত্ত রাশি = 14a² – 29ab – 15b²
= 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

সমাধান: (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 + (pq + 1)xy + qy²
= 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

সমাধান: 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)

২২. a³ – 6a² + 12a – 9

সমাধান: 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)

২৩. a³-9b³+(a+b)³

সমাধান: 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)

২৪. 8x³ + 12x² + 6x – 63

সমাধান: 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)

২৫. 8a³ + b³/27

সমাধান: 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

২৬. a⁶/27-b⁶

সমাধান: 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)

২৭. 4a² + 1/4a² – 2 + 4a – 1/a

সমাধান: 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)

২৮. (3a + 1)³ – (2a – 3)³

সমাধান: (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)

২৯. (x + 2)(x + 3)(x + 4)(x + 5) – 48

সমাধান: (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 = m(m + 2) – 48
= 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

সমাধান:(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 = p(p + 8) – 65
= 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)

সমাধান:

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)