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# Koshe Dekhi 8.2Class 9

### Koshe Dekhi 8.2 Class 9

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

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

সমাধানঃ

$\fn_cm&space;\frac{x^{4}}{16}-\frac{y^{4}}{81}$

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

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

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

2. $\fn_cm&space;{\color{Blue}&space;m^{2}+\frac{1}{m^{2}}+2-2m-\frac{2}{m}}$

সমাধানঃ

$\fn_cm&space;m^{2}+\frac{1}{m^{2}}+2-2m-\frac{2}{m}$

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

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

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

$=\left&space;(&space;m+\frac{1}{m}&space;\right&space;)^{2}-2+2-2\left&space;(&space;m+\frac{1}{m}&space;\right&space;)$

$=\left&space;(&space;m+\frac{1}{m}&space;\right&space;)^{2}-2\left&space;(&space;m+\frac{1}{m}&space;\right&space;)$

$=\left&space;(&space;m+\frac{1}{m}&space;\right&space;)\left&space;(&space;m+\frac{1}{m}-2&space;\right&space;)$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

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

সমাধানঃ

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

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

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

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

$=\left&space;(&space;3p-4q&space;\right&space;)\left&space;(&space;3p-4q&space;\right&space;)+a\left&space;(&space;3p-4q&space;\right&space;)$

$=\left&space;(&space;3p-4q&space;\right&space;)\left&space;[\left&space;(&space;3p-4q&space;\right&space;)+a&space;\right&space;]$

$=\left&space;(&space;3p-4q&space;\right&space;)\left&space;(&space;3p-4q+a&space;\right&space;)$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

4. $\fn_cm&space;{\color{Blue}&space;4x^{4}+81}$

সমাধানঃ

$\fn_cm&space;4x^{4}+81$

$=\left&space;(&space;2x^{2}&space;\right&space;)^{2}+\left&space;(&space;9&space;\right&space;)^{2}$

$=\left&space;(&space;2x^{2}+9&space;\right&space;)^{2}-2\times&space;2x^{2}\times&space;9$

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

$=\left&space;(&space;2x^{2}+9&space;\right&space;)^{2}-36x^{2}$

$=\left&space;(&space;2x^{2}+9&space;\right&space;)^{2}-\left&space;(6x&space;\right&space;)^{2}$

$=\left&space;(&space;2x^{2}+9+6x&space;\right&space;)\left&space;(&space;2x^{2}+9-6x&space;\right&space;)$

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

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

সমাধানঃ

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

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

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

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

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

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

$=\left&space;(&space;x^{2}+1+3x&space;\right&space;)\left&space;(&space;x^{2}+1-3x&space;\right&space;)$

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

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

সমাধানঃ

$\fn_cm&space;p^{4}-11p^{2}q^{2}+q^{4}$

$=\left&space;(&space;p^{2}&space;\right&space;)^{2}-2p^{2}q^{2}+\left&space;(&space;q^{2}&space;\right&space;)^{2}-9p^{2}q^{2}$

$=\left&space;(&space;p^{2}-q^{2}&space;\right&space;)^{2}-9p^{2}q^{2}$

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

$=\left&space;(&space;p^{2}-q^{2}&space;\right&space;)^{2}-\left&space;(3pq&space;\right&space;)^{2}$

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

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

সমাধানঃ

$\fn_cm&space;a^{2}+b^{2}-c^{2}-2ab$

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

$=\left&space;(&space;a-b&space;\right&space;)^{2}-\left&space;(&space;c&space;\right&space;)^{2}$

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

$=\left&space;(&space;a-b+c&space;\right&space;)\left&space;(&space;a-b-c&space;\right&space;)$

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

(উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

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

সমাধানঃ

$\fn_cm&space;3a\left&space;(&space;3a+2c&space;\right&space;)-4b\left&space;(&space;b+c&space;\right&space;)$

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

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

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

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

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

$=\left&space;(&space;3a+c+2b+c&space;\right&space;)\left&space;(&space;3a+c-2b-c&space;\right&space;)$

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

$=\left&space;(&space;3a+2b+2c&space;\right&space;)\left&space;(&space;3a-2b&space;\right&space;)$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

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

সমাধানঃ

$\fn_cm&space;a^{2}-6ab+12bc-4c^{2}$

$=a^{2}-2\times&space;a\times&space;3b+2\times&space;3b\times&space;2c-4c^{2}$

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

$=\left&space;(&space;a-3b&space;\right&space;)^{2}-\left&space;(&space;3b-2c&space;\right&space;)^{2}$

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

$=\left&space;(&space;a-3b+3b-2c&space;\right&space;)\left&space;(&space;a-3b-3b+2c&space;\right&space;)$

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

$=\left&space;(&space;a-2c&space;\right&space;)\left&space;(&space;a-6b+2c&space;\right&space;)$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

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

সমাধানঃ

$\fn_cm&space;3a^{2}+4ab+b^{2}-2ac-c^{2}$

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

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

$=\left&space;(&space;2a+b&space;\right&space;)^{2}-\left&space;(&space;a+c&space;\right&space;)^{2}$

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

$=\left&space;(&space;2a+b+a+c&space;\right&space;)\left&space;(&space;2a+b-a-c&space;\right&space;)$

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

$=\left&space;(&space;3a+b+c&space;\right&space;)\left&space;(&space;a+b-c&space;\right&space;)$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

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

সমাধানঃ

$\fn_cm&space;x^{2}-y^{2}-6ax+2ay+8a^{2}$

$=x^{2}-y^{2}-2\times&space;x\times&space;3a+2\times&space;a\times&space;y+9a^{2}-a^{2}$

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

$=\left&space;(&space;x-3a&space;\right&space;)^{2}-\left&space;(&space;a-y&space;\right&space;)^{2}$

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

$=\left&space;(&space;x-3a+a-y&space;\right&space;)\left&space;(&space;x-3a-a+y&space;\right&space;)$

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

$=\left&space;(&space;x-y-2a&space;\right&space;)\left&space;(&space;x+y-4a&space;\right&space;)$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

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

সমাধানঃ

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

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

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

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

$=\left&space;(&space;a-2c+3b-5d&space;\right&space;)\left&space;(&space;a-2c-3b+5d&space;\right&space;)$

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

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

সমাধানঃ

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

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

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

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

$=\left&space;(&space;2a&space;\right&space;)^{2}-\left&space;(&space;-a+b+c&space;\right&space;)^{2}$

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

$=\left&space;[&space;\left&space;(&space;2a&space;\right&space;)+\left&space;(&space;-a+b+c&space;\right&space;)&space;\right&space;]\left&space;[&space;\left&space;(&space;2a&space;\right&space;)-\left&space;(&space;-a+b+c&space;\right&space;)&space;\right&space;]$

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

$=\left&space;(&space;2a-a+b+c&space;\right&space;)\left&space;(&space;2a+a-b-c&space;\right&space;)$

$=\left&space;(&space;a+b+c&space;\right&space;)\left&space;(&space;3a-b-c&space;\right&space;)$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

14. $\fn_cm&space;{\color{Blue}&space;x^{2}-2x-22499}$

সমাধানঃ

$\fn_cm&space;x^{2}-2x-22499$

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

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

$=\left&space;(&space;x-1&space;\right&space;)^{2}-\left&space;(&space;150&space;\right&space;)^{2}$

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

$=\left&space;(&space;x-1+150&space;\right&space;)\left&space;(&space;x-1-150&space;\right&space;)$

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

$=\left&space;(x+149&space;\right&space;)\left&space;(&space;x-151&space;\right&space;)$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.2

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

সমাধানঃ

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

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

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

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

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

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

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

$=\left&space;(ax+by+bx-ay&space;\right&space;)\left&space;(&space;ax+by-bx+ay&space;\right&space;)$

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

$=\left&space;(&space;ax-bx+ay+by\right&space;)\left&space;(ax+bx-ay+by&space;\right&space;)$ (উত্তর)

### Koshe dekhi 8.2 Class 9

Thank You

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