Wed. Apr 24th, 2024

Koshe Dekhi 8.2 Class 9

Koshe Dekhi 8.2 Class 9

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

1. \fn_cm {\color{Blue} \frac{x^{4}}{16}-\frac{y^{4}}{81}}

সমাধানঃ 

\fn_cm \frac{x^{4}}{16}-\frac{y^{4}}{81}

=\left ( \frac{x^{2}}{4} \right )^{2}-\left ( \frac{y^{2}}{9} \right )^{2}

=\left [ \left ( \frac{x}{2} \right )^{2} \right ]^{2}-\left [ \left ( \frac{y}{3} \right )^{2} \right ]^{2}

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

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

=\left [\left ( \frac{x^{2}}{4} \right )+\left ( \frac{y^{2}}{9} \right ) \right ]\left [ \left (\frac{x}{2} + \frac{y}{3} \right ) \left (\frac{x}{2} - \frac{y}{3} \right ) \right ]

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

=\left ( \frac{x^{2}}{4}+\frac{y^{2}}{9} \right ) \left (\frac{x}{2} + \frac{y}{3} \right ) \left (\frac{x}{2} - \frac{y}{3} \right ) (উত্তর)

 

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

2. \fn_cm {\color{Blue} m^{2}+\frac{1}{m^{2}}+2-2m-\frac{2}{m}}

সমাধানঃ 

\fn_cm m^{2}+\frac{1}{m^{2}}+2-2m-\frac{2}{m}

\fn_cm =\left (m \right )^{2}+\left (\frac{1}{m} \right )^{2}+2-2\left (m+\frac{1}{m} \right )

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

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

=\left ( m+\frac{1}{m} \right )^{2}-2+2-2\left ( m+\frac{1}{m} \right )

=\left ( m+\frac{1}{m} \right )^{2}-2\left ( m+\frac{1}{m} \right )

=\left ( m+\frac{1}{m} \right )\left ( m+\frac{1}{m}-2 \right ) (উত্তর)

 

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

3. \fn_cm {\color{Blue} 9p^{2}-24pq+16q^{2}+3ap-4aq}

সমাধানঃ 

9p^{2}-24pq+16q^{2}+3ap-4aq

=\left [\left (3p \right )^{2}-2\times 3p\times 4q+\left (4q \right )^{2} \right ]+a\left (3p-4q \right )

=\left ( 3p-4q \right )^{2}+a\left ( 3p-4q \right )

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

=\left ( 3p-4q \right )\left ( 3p-4q \right )+a\left ( 3p-4q \right )

=\left ( 3p-4q \right )\left [\left ( 3p-4q \right )+a \right ]

=\left ( 3p-4q \right )\left ( 3p-4q+a \right ) (উত্তর)

 

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

4. \fn_cm {\color{Blue} 4x^{4}+81}

সমাধানঃ 

\fn_cm 4x^{4}+81

=\left ( 2x^{2} \right )^{2}+\left ( 9 \right )^{2}

=\left ( 2x^{2}+9 \right )^{2}-2\times 2x^{2}\times 9

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

=\left ( 2x^{2}+9 \right )^{2}-36x^{2}

=\left ( 2x^{2}+9 \right )^{2}-\left (6x \right )^{2}

=\left ( 2x^{2}+9+6x \right )\left ( 2x^{2}+9-6x \right )

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

=\left ( 2x^{2}+6x+9 \right )\left ( 2x^{2}-6x+9 \right ) (উত্তর)

 

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

5. \fn_cm {\color{Blue} x^{4}-7x^{2}+1}

সমাধানঃ 

x^{4}-7x^{2}+1

=x^{4}+2x^{2}+1-9x^{2}

=\left [\left (x^{2} \right )^{2}+2x^{2}+1 \right ]-9x^{2}

=\left [\left (x^{2} \right )^{2}+2\times x^{2}\times 1+\left ( 1 \right )^{2} \right ]-9x^{2}

=\left [\left (x^{2}+1 \right ) \right ]^{2}-\left (3x \right )^{2}

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

=\left ( x^{2}+1+3x \right )\left ( x^{2}+1-3x \right )

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

=\left ( x^{2}+3x+1 \right )\left ( x^{2}-3x+1 \right ) (উত্তর)

 

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

6. \fn_cm {\color{Blue} p^{4}-11p^{2}q^{2}+q^{4}}

সমাধানঃ 

\fn_cm p^{4}-11p^{2}q^{2}+q^{4}

=\left ( p^{2} \right )^{2}-2p^{2}q^{2}+\left ( q^{2} \right )^{2}-9p^{2}q^{2}

=\left ( p^{2}-q^{2} \right )^{2}-9p^{2}q^{2}

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

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

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

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

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

 

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

7. \fn_cm {\color{Blue} a^{2}+b^{2}-c^{2}-2ab}

সমাধানঃ 

\fn_cm a^{2}+b^{2}-c^{2}-2ab

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

=\left ( a-b \right )^{2}-\left ( c \right )^{2}

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

=\left ( a-b+c \right )\left ( a-b-c \right )

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

(উত্তর)

 

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

8. \fn_cm {\color{Blue} 3a\left ( 3a+2c \right )-4b\left ( b+c \right )}

সমাধানঃ 

\fn_cm 3a\left ( 3a+2c \right )-4b\left ( b+c \right )

=9a^{2}+6ac-4b^{2}-4bc

=9a^{2}+6ac+c^{2}-4b^{2}-4bc-c^{2}

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

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

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

=\left ( 3a+c+2b+c \right )\left ( 3a+c-2b-c \right )

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

=\left ( 3a+2b+2c \right )\left ( 3a-2b \right ) (উত্তর)

 

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

9. \fn_cm {\color{Blue} a^{2}-6ab+12bc-4c^{2}}

সমাধানঃ 

\fn_cm a^{2}-6ab+12bc-4c^{2}

=a^{2}-2\times a\times 3b+2\times 3b\times 2c-4c^{2}

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

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

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

=\left ( a-3b+3b-2c \right )\left ( a-3b-3b+2c \right )

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

=\left ( a-2c \right )\left ( a-6b+2c \right ) (উত্তর)

 

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

10. \fn_cm {\color{Blue} 3a^{2}+4ab+b^{2}-2ac-c^{2}}

সমাধানঃ 

\fn_cm 3a^{2}+4ab+b^{2}-2ac-c^{2}

=4a^{2}+4ab+b^{2}-a^{2}-2ac-c^{2}

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

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

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

=\left ( 2a+b+a+c \right )\left ( 2a+b-a-c \right )

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

=\left ( 3a+b+c \right )\left ( a+b-c \right ) (উত্তর)

 

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

11. \fn_cm {\color{Blue} x^{2}-y^{2}-6ax+2ay+8a^{2}}

সমাধানঃ 

\fn_cm x^{2}-y^{2}-6ax+2ay+8a^{2}

=x^{2}-y^{2}-2\times x\times 3a+2\times a\times y+9a^{2}-a^{2}

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

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

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

=\left ( x-3a+a-y \right )\left ( x-3a-a+y \right )

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

=\left ( x-y-2a \right )\left ( x+y-4a \right ) (উত্তর)

 

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

12. \fn_cm {\color{Blue} a^{2}-9b^{2}+4c^{2}-25d^{2}-4ac+30bd}

সমাধানঃ 

\fn_cm a^{2}-9b^{2}+4c^{2}-25d^{2}-4ac+30bd

=\left [a^{2}-2\times a\times 2c+\left ( 2c \right )^{2} \right ]-\left [ \left ( 3b \right )^{2}-2\times 3b\times 5d+\left ( 5d \right )^{2} \right ]

=\left ( a-2c \right )^{2}-\left ( 3b-5d \right )^{2}

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

=\left ( a-2c+3b-5d \right )\left ( a-2c-3b+5d \right )

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

=\left ( a+3b-2c-5d \right )\left ( a-3b-2c+5d \right ) (উত্তর)

 

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

13. \fn_cm {\color{Blue} 3a^{2}-b^{2}-c^{2}+2ab-2bc+2ca}

সমাধানঃ 

\fn_cm 3a^{2}-b^{2}-c^{2}+2ab-2bc+2ca

=4a^{2}-a^{2}-b^{2}-c^{2}+2ab-2bc+2ca

=\left (2a \right )^{2}-\left (a^{2}+b^{2}+c^{2}-2ab+2bc-2ca \right )

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

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

[ সূত্রঃ {\color{Blue} x^{2}+y^{2}+z^{2}+2xy+2yz+2zx=\left ( x+y+z \right )^{2}} ]

=\left [ \left ( 2a \right )+\left ( -a+b+c \right ) \right ]\left [ \left ( 2a \right )-\left ( -a+b+c \right ) \right ]

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

=\left ( 2a-a+b+c \right )\left ( 2a+a-b-c \right )

=\left ( a+b+c \right )\left ( 3a-b-c \right ) (উত্তর)

 

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

14. \fn_cm {\color{Blue} x^{2}-2x-22499}

সমাধানঃ 

\fn_cm x^{2}-2x-22499

=x^{2}-2x+1-22500

=\left ( x^{2}-2.x.1+1^{2} \right )-\left ( 150 \right )^{2}

=\left ( x-1 \right )^{2}-\left ( 150 \right )^{2}

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

=\left ( x-1+150 \right )\left ( x-1-150 \right )

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

=\left (x+149 \right )\left ( x-151 \right ) (উত্তর)

 

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

15. \fn_cm {\color{Blue} \left ( x^{2}-y^{2} \right )\left ( a^{2}-b^{2} \right )+4abxy}

সমাধানঃ 

\fn_cm \left ( x^{2}-y^{2} \right )\left ( a^{2}-b^{2} \right )+4abxy

=a^{2}x^{2}-b^{2}x^{2}-a^{2}y^{2}+b^{2}y^{2}+4abxy

=a^{2}x^{2}-b^{2}x^{2}-a^{2}y^{2}+b^{2}y^{2}+2abxy+2abxy

=a^{2}x^{2}+2abxy+b^{2}y^{2}-\left [b^{2}x^{2}-2abxy+a^{2}y^{2} \right ]

=\left ( ax+by \right )^{2}-\left ( bx-ay \right )^{2}

[ সূত্রঃ {\color{Blue} m^{2}+2mn+n^{2}=\left ( m+n \right )^{2}}  ও

{\color{Blue} m^{2}-2mn+n^{2}=\left ( m-n \right )^{2}} ]

=\left (ax+by+bx-ay \right )\left ( ax+by-bx+ay \right )

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

=\left ( ax-bx+ay+by\right )\left (ax+bx-ay+by \right ) (উত্তর)

Koshe dekhi 8.2 Class 9

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