Fri. May 24th, 2024

Dec 9, 2020

Koshe Dekhi 8.3Class 9

Koshe Dekhi 8.3 Class 9

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

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

সমাধানঃ

$\fn_cm&space;t^{9}-512$

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

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

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

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

সমাধানঃ

$\fn_cm&space;729p^{6}-q^{6}$

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

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

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

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

সমাধানঃ

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

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

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

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

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

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

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

সমাধানঃ

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

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

সমাধানঃ

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

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

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

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

সমাধানঃ

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

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

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

সমাধানঃ

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

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

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

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

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

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

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

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

(উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

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

সমাধানঃ

$\fn_cm&space;32x^{4}-500x$

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

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

সমাধানঃ

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

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

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.3

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

সমাধানঃ

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

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

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

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

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

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

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

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

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

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

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

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

Koshe dekhi 8.3 Class 9

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