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

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

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

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

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