Fri. May 24th, 2024

Koshe Dekhi 8.3 Class 9

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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