Koshe Dekhi 8.3 Class 9

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

1. $\fn_cm {\color{Blue} t^{9}-512}$

সমাধানঃ

$\fn_cm t^{9}-512$

$=\left ( t^{3} \right )^{3}-\left ( 8 \right )^{3}$

$=\left ( t^{3}-8 \right )\left [\left ( t^{3}\right )^{2}+t^{3}\times 8 +\left ( 8 \right )^{2} \right ]$

[ সূত্রঃ ${\color{Blue} a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2}+ab+b^{2} \right )}$ ]

$=\left ( t^{3}-2^{3} \right )\left [t^{6}+8t^{3}+64\right ]$

$=\left [\left ( t-2 \right )\left ( t^{2} +t\times 2+2^{2}\right ) \right ]\left [ t^{6}+8t^{3}+64 \right ]$

[ সূত্রঃ ${\color{Blue}a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2}+ab+b^{2} \right )}$ ]

$=\left ( t-2 \right )\left ( t^{2}+2t+4 \right )\left (t^{6}+8t^{3}+64\right )$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

2. $\fn_cm {\color{Blue} 729p^{6}-q^{6}}$

সমাধানঃ

$\fn_cm 729p^{6}-q^{6}$

$=\left ( 27p^{3} \right )^{2}-\left ( q^{3} \right )^{2}$

$=\left ( 27p^{3}+q^{3} \right )\left ( 27p^{3}-q^{3} \right )$

[ সূত্রঃ ${\color{Blue} a^{2}-b^{2}=\left ( a+b \right )\left ( a-b \right )}$ ]

$=\left [ \left ( 3p \right )^{3}+\left ( q \right )^{3}\right ]\left [ \left ( 3p \right )^{3}-\left ( q \right )^{3} \right ]$

$=\left [\left ( 3p+q \right )\left ( 9p^{2}-3pq+q^{2} \right ) \right ]\left [\left ( 3p-q \right )\left ( 9p^{2}+3pq+q^{2} \right ) \right ]$

[ সূত্রঃ ${\color{Blue}a^{3}+b^{3}=\left ( a+b \right )\left ( a^{2}-ab+b^{2} \right ); a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2}+ab+b^{2} \right )}$ ]

$=\left ( 3p+q \right )\left ( 3p-q \right )\left ( 9p^{2}-3pq+q^{2} \right ) \left ( 9p^{2}+3pq+q^{2} \right )$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

3. $\fn_cm {\color{Blue} 8\left ( p-3 \right )^{3}+343}$

সমাধানঃ

$\fn_cm 8\left ( p-3 \right )^{3}+343$

$\fn_cm =\left [\left ( 2 \right )^{3}\times \left ( p-3 \right )^{3} \right ]+\left ( 7 \right )^{3}$

$\fn_cm =\left [2\left ( p-3 \right )\right ]^{3} +\left ( 7 \right )^{3}$

$\fn_cm =\left ( 2p-6 \right )^{3}+\left ( 7 \right )^{3}$

$=\left ( 2p-6+7 \right )\left [ \left ( 2p-6 \right )^{2}-\left ( 2p-6 \right )\times 7+\left ( 7 \right )^{2} \right ]$

[ সূত্রঃ ${\color{Blue}a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2}+ab+b^{2} \right )}$ ]

$=\left ( 2p+1 \right )\left [ \left ( 2p \right )^{2}-2\times 2p\times 6+\left (6 \right )^{2}-2p\times 7-\left ( -6 \right )\times 7+49 \right ]$

[ সূত্রঃ ${\color{Blue} \left ( a-b \right )^{2}=a^{2}-2ab+b^{2}}$ ]

$=\left ( 2p+1 \right )\left [ 4p^{2}-24p+36-14p+42+49 \right ]$

$=\left ( 2p+1 \right )\left ( 4p^{2}-38p+127 \right )$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

4. $\fn_cm {\color{Blue} \frac{1}{8a^{3}}+\frac{8}{b^{3}}}$

সমাধানঃ

$\fn_cm \frac{1}{8a^{3}}+\frac{8}{b^{3}}$

$=\left ( \frac{1}{2a} \right )^{3}+\left ( \frac{2}{b} \right )^{3}$

[ সূত্রঃ ${\color{Blue} x^{3}+y^{3}=\left ( x+y \right )\left ( x^{2}-xy+y^{2} \right )}$ ]

$=\left ( \frac{1}{2a}+\frac{2}{b} \right )\left [ \left ( \frac{1}{2a} \right )^{2}-\frac{1}{2a}\times \frac{2}{b}+\left ( \frac{2}{b} \right )^{2} \right ]$

$=\left ( \frac{1}{2a}+\frac{2}{b} \right )\left ( \frac{1}{4a^{2}}-\frac{1}{ab}+\frac{4}{b^{2}} \right )$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

5. $\fn_cm {\color{Blue} \left ( 2a^{3}-b^{3} \right )^{3}-b^{9}}$

সমাধানঃ

$\fn_cm \left ( 2a^{3}-b^{3} \right )^{3}-b^{9}$

$\fn_cm =\left ( 2a^{3}-b^{3} \right )^{3}-\left ( b^{3} \right )^{3}$

$=\left ( 2a^{3}-b^{3}-b^{3} \right )\left [ \left ( 2a^{3}-b^{3} \right )^{2}+\left (2a^{3}-b^{3} \right )b^{3}+\left ( b^{3} \right )^{2} \right ]$

[ সূত্রঃ ${\color{Blue} x^{3}-y^{3}=\left ( x-y \right )\left ( x^{2}+xy+y^{2} \right )}$ ]

$=\left ( 2a^{3}-2b^{3} \right )\left [ 4a^{6} -4a^{3}b^{3}+b^{6}+2a^{3}b^{3}-b^{6}+b^{6} \right ]$

$=2\left ( a^{3}-b^{3} \right )\left ( 4a^{6} -2a^{3}b^{3}+b^{6} \right )$

$=2\left ( a-b \right )\left ( a^{2}+ab+b^{2} \right )\left ( 4a^{6} -2a^{3}b^{3}+b^{6} \right )$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

6. $\fn_cm {\color{Blue} AR^{3}-Ar^{3}+AR^{2}h-Ar^{2}h}$

সমাধানঃ

$\fn_cm AR^{3}-Ar^{3}+AR^{2}h-Ar^{2}h$

$\fn_cm =A\left (R^{3}-r^{3} \right )+Ah\left (R^{2}-r^{2} \right )$

$=A\left [ \left ( R-r \right )\left ( R^{2}+Rr+r^{2} \right ) \right ]+Ah\left [ \left ( R+r \right )\left ( R-r \right ) \right ]$

[ সূত্রঃ ${\color{Blue}a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2}+ab+b^{2} \right );a^{2}-b^{2}=\left ( a+b \right )\left ( a-b \right )}$ ]

$=A\left ( R-r \right )\left [\left ( R^{2}+Rr+r^{2} \right )+h\left ( R+r \right ) \right ]$

$=A\left ( R-r \right ) \left (R^{2}+Rr+r^{2} +Rh+hr \right )$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

7. $\fn_cm {\color{Blue} a^{3}+3a^{2}b+3ab^{2}+b^{3}-8}$

সমাধানঃ

$\fn_cm a^{3}+3a^{2}b+3ab^{2}+b^{3}-8$

$\fn_cm =\left (a^{3}+3a^{2}b+3ab^{2}+b^{3} \right )-8$

$=\left ( a+b \right )^{3}-\left ( 2 \right )^{3}$

[ সূত্রঃ $\fn_cm {\color{Blue} \left (a^{3}+3a^{2}b+3ab^{2}+b^{3} \right )=\left ( a+b \right )^{3}}$ ]

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

[ সূত্রঃ ${\color{Blue} x^{3}-y^{3}=\left ( x-y \right )\left ( x^{2}+xy+y^{2} \right )}$ ]

$=\left ( a+b-2 \right )\left ( a^{2}+2ab+b^{2}+2a+2b+4 \right )$

[ সূত্রঃ ${\color{Blue} \left ( a+b \right )^{2}=a^{2}+2ab+b^{2} }$ ]

(উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

8. $\fn_cm {\color{Blue} 32x^{4}-500x}$

সমাধানঃ

$\fn_cm 32x^{4}-500x$

$\fn_cm =4x\left (8x^{3}-125 \right )$

$=4x\left [\left ( 2x \right )^{3}-\left ( 5 \right )^{3} \right ]$

$=4x\left [ \left ( 2x-5 \right )\left \{ \left ( 2x \right )^{2}+2x\times 5+5^{2} \right \} \right ]$

[ সূত্রঃ ${\color{Blue} a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2}+ab+b^{2} \right )}$ ]

$=4x\left [ \left ( 2x-5 \right )\left (4x^{2}+10x+25 \right ) \right ]$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

9. $\fn_cm {\color{Blue} 8a^{3}-b^{3}-4ax+2bx}$

সমাধানঃ

$\fn_cm 8a^{3}-b^{3}-4ax+2bx$

$\fn_cm =\left [8a^{3}-b^{3} \right ]-\left [4ax-2bx \right ]$

$\fn_cm =\left [\left (2a \right )^{3}-\left (b \right )^{3} \right ]-\left [2x\left ( 2a-b \right ) \right ]$

$\fn_cm =\left [\left (2a-b \right )\left ( 4a^{2}+2ab+b^{2} \right ) \right ]-\left [2x\left ( 2a-b \right ) \right ]$

[ সূত্রঃ ${\color{Blue} a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2}+ab+b^{2} \right )}$ ]

$\fn_cm =\left (2a-b \right )\left [\left ( 4a^{2}+2ab+b^{2} \right )-2x \right ]$

$\fn_cm =\left (2a-b \right )\left ( 4a^{2}+2ab+b^{2}-2x \right )$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

10. $\fn_cm {\color{Blue} x^{3}-6x^{2}+12x-35}$

সমাধানঃ

$\fn_cm x^{3}-6x^{2}+12x-35$

$\fn_cm =x^{3}-6x^{2}+12x-35-8+8$

$\fn_cm =\left [x^{3}-6x^{2}+12x-8 \right ]+8-35$

$=\left [\left ( x \right )^{3}-3\times \left ( x \right )^{2}\times 2+3\times x\times \left ( 2 \right )^{2}-\left ( 2 \right )^{3} \right ]-27$

$=\left [\left ( x-2 \right )^{3} \right ]-27$

[ সূত্রঃ ${\color{Blue} a^{3}-3a^{2}b+3ab^{2}-b^{3}=\left ( a-b \right )^{3}}$ ]

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

$=\left ( x-2-3 \right )\left [ \left ( x-2 \right )^{2}+\left ( x-2 \right )\times 3+\left ( 3 \right )^{2} \right ]$

[ সূত্রঃ ${\color{Blue}a^{3}-b^{3}=\left ( a-b \right )\left ( a^{2}+ab+b^{2} \right )}$ ]

$=\left ( x-5 \right )\left [ x^{2}-4x+4+3x-6+9 \right ]$

[ সূত্রঃ ${\color{Blue} \left ( a-b \right )^{2}=a^{2}-2ab+b^{2}}$ ]

$=\left ( x-5 \right )\left ( x^{2}-x+7 \right )$ (উত্তর)

Koshe dekhi 8.3 Class 9

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Koshe dekhi 8.3 class 9

Koshe Dekhi 8.3 Class 9

Koshe Dekhi 8.3 Class 9

Koshe dekhi 8.3 Class 9

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