Class IX - Factorisation
Math
Exercise 8.3
নবম শ্রেণী
অংক
কষে দেখি - 8.3
উৎপাদকে বিশ্লেষণ
1. t⁹ - 512
2. 729p⁶ - q⁶
3. 8(p - 3)³ + 343
4. $\frac{1}{8a³}$ + $\frac{8}{b³}$
5. (2a³ - b³)³ – b⁹
6. AR³ - Ar³ + AR²h – Ar²h
7. a³ + 3a²b + 3ab² + b³ – 8
8. 32x⁴ – 500x
9. 8a³ - b³ - 4ax + 2bx
10. x³ - 6x² + 12x - 35
Solution
1. t⁹ - 512
Ans.
t⁹ - 512
= (t³)³ - (8)³
= (t³ - 8){(t³)² + t³.8 + 8²}
= (t³ - 2³)(t⁶ + 8t³ + 64)
= (t - 2)(t² + t.2 + 2²)(t⁶ + 8t³ + 64)
= (t - 2)(t² + 2t + 4)(t⁶ + 8t³ + 64)
উত্তর: নির্ণয় উৎপাদন (t - 2)(t² + 2t +4)(t⁶ + 8t³ + 64)
2. 729p⁶ - q⁶
Ans.
729p⁶ - q⁶
= (27p³)² - (q³)²
= (27p³ + q³) (27p³ - q³)
= {(3p)³ + (q)³}{(3p)³ - (q)³}
= (3p + q){(3p)² - 3p.q + q²} (3p - q){(3p)² + 3p.q + q²}
= (3p + q)(9p² - 3pq + q²)(3p - q)(9p² + 3pq + q²)
= (3p + q)(3p - q)(9p² - 3pq + q²)(9p² + 3pq + q²)
উত্তর: নির্ণয় উৎপাদন (3p + q)(3p - q)(9p² - 3pq + q²)(9p² + 3pq + q²)
3. 8(p - 3)³ + 343
Ans.
8(p - 3)³ + 343
= {2(p - 3)}³ + (7)³
= {2(p - 3) + 7}[{2(p - 3)}² - 2(p - 3).7 + 7²}
= (2p - 3 + 7){4(p² - 6p + 9) - 14p + 42 + 49}
= (2p + 1)(4p² - 24p + 36 - 14p + 91)
= (2p + 1)(4p² - 38p + 127)
উত্তর: নির্ণয় উৎপাদন (2p + 1)(4p² - 38p + 127)
4. $\frac{1}{8a³}$ + $\frac{8}{b³}$
Ans.
$\frac{1}{8a³}$ + $\frac{8}{b³}$
= $(\frac{1}{2a})^3$ + $(\frac{2}{b})^3$
= $(\frac{1}{2a}$ + $\frac{2}{b})$ {$(\frac{1}{2a})^2$ - $\frac{1}{2a}$$\frac{2}{b}$ + $(\frac{2}{b})^2$}
= $(\frac{1}{2a}$ + $\frac{2}{b})$ $(\frac{1}{4a²}$ - $\frac{1}{ab}$ + $\frac{4}{b²})$
উত্তর: নির্ণয় উৎপাদন $(\frac{1}{2a}$ + $\frac{2}{b})$ $(\frac{1}{4a²}$ - $\frac{1}{ab}$ + $\frac{4}{b²})$
5. (2a³ - b³)³ – b⁹
Ans.
(2a³ - b³)³ – b⁹
= (2a³ - b³)³ - (b³)³
= {(2a³ - b³) - b³}{(2a³ - b³)² + (2a³ - b³).b³ + (b³)²}
= (2a³ - b³ - b³)(4a⁶ - 4a³b³ + b⁶ + 2a³b³ - b⁶ + b⁶}
= (2a³ - 2b³)(4a⁶ - 2a³b³ + b⁶}
= 2(a³ - b³)(4a⁶ - 2a³b³ + b⁶}
= 2(a - b)(a² + ab + b²)(4a⁶ - 2a³b³ + b⁶}
উত্তর: নির্ণয় উৎপাদন 2(a - b)(a² + ab + b²)(4a⁶ - 2a³b³ + b⁶)
6. AR³ - Ar³ + AR²h – Ar²h
Ans.
AR³ - Ar³ + AR²h – Ar²h
= A(R³ - r³) + Ah(R² - r²)
= A(R - r)(R² + Rr + r²) + Ah(R + r)(R - r)
= A(R - r){(R² + Rr + r²) + h(R + r)}
= A(R - r)(R² + Rr + r² + Rh + rh)
উত্তর: নির্ণয় উৎপাদন A(R - r)(R² + Rr + r² + Rh + rh)
7. a³ + 3a²b + 3ab² + b³ – 8
Ans.
a³ + 3a²b + 3ab² + b³ – 8
= (a + b)³ - (2)³
= {(a + b) - 2}{(a + b)² + (a + b).2 + 2²}
= (a + b - 2)(a² + 2ab + b² + 2a + 2b + 4}
উত্তর: নির্ণয় উৎপাদন (a + b - 2)(a² + 2ab + b² + 2a + 2b + 4}
8. 32x⁴ – 500x
Ans.
32x⁴ – 500x
= 4x(8x³ - 125)
= 4x{(2x)³ - (5)³}
= 4x(2x - 5){(2x)² +2x.5 + (5)²}
= 4x(2x - 5)(4x² +10x + 25)
উত্তর: নির্ণয় উৎপাদন 4x(2x - 5)(4x² +10x + 25)
9. 8a³ - b³ - 4ax + 2bx
Ans.
8a³ - b³ - 4ax + 2bx
= (2a)³ - b³ - 4ax + 2bx
= (2a - b){(2a)² + 2a.b + b²} - 2x(2a - b)
= (2a - b)(4a² + 2ab + b²) - 2x(2a - b)
= (2a - b)(4a² + 2ab + b² - 2x)
উত্তর: নির্ণয় উৎপাদন (2a - b)(4a² + 2ab + b² - 2x)
10. x³ - 6x² + 12x - 35
Ans.
x³ - 6x² + 12x - 35
= x³ - 3.x².2 + 3.x.4 - 8 - 27
= x³ - 3.x².2 + 3.x.2² - 2³ - 27
= (x - 2)³ - (3)³
= {(x - 2) - (3)}{(x - 2)² + (x - 2).(3) + (3)²}
= (x - 2 - 3)(x² - 4x + 4 + 3x - 6 + 9)
= (x - 5)(x² - x + 7)
উত্তর: নির্ণয় উৎপাদন (x - 5) (x² - x+7)
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