৮ম শ্রেণির অনুশীলনী ৪.৩ — উৎপাদকে বিশ্লেষণ
\begin{align*}
&\mathbf{1. a^{3}+8}\\
&সমাধানঃ\\
&a^{3}+8\\
&=a^{3}+2^{3}\\
&= \left(a+2\right)\left(a^{2}-a.2+2^{2}\right) [ সূত্র a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right) ]\\
&=\left(a+2\right)\left(a^{2}-2a+4\right)\\
&\mathbf{2. 8x^{3}+343}\\
&সমাধানঃ\\
&8x^{3}+343\\
&={\left(2x\right)}^{3}+7^{3}\\
&= \left(2x+7\right)\left\{{\left(2x\right)}^{2}-2x.7+7^{2}\right\} [ সূত্র a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right) ]\\
&=\left(2x+7\right)\left(4x^{2}-14x+49\right)\\
&\mathbf{3. 8a^{4}+27ab^{3}}\\
&সমাধানঃ\\
&8a^{4}+27ab^{3}\\
&=a\left(8a^{3}+27b^{3}\right) [কমন]\\
&=a\left\{{\left(2a\right)}^{3}+{\left(3b\right)}^{3}\right\}\\
&= a\left(2a+3b\right)\left\{{\left(2a\right)}^{2}-2a.3b+{\left(3b\right)}^{2}\right\} [ সূত্র a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right) ]\\
&=a\left(2a+3b\right)\left(4a^{2}-6ab+9b^{2}\right)\\
&\mathbf{4. 8x^{3}+1}\\
&সমাধানঃ\\
&8x^{3}+1\\
&={\left(2x\right)}^{3}+1^{3}\\
&= \left(2x+1\right)\left\{{\left(2x\right)}^{2}-2x.1+1^{2}\right\} [ সূত্র a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right) ]\\
&=\left(2x+1\right)\left(4x^{2}-2x+1\right)\\
&\mathbf{5. 64a^{3}-125b^{3}}\\
&সমাধানঃ\\
&64a^{3}-125b^{3}\\
&={\left(4a\right)}^{3}-{\left(5b\right)}^{3}\\
&= \left(4a-5b\right)\left\{{\left(4a\right)}^{2}+4a.5b+{\left(5b\right)}^{2}\right\} [ সূত্র a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+\ ab+b^{2}\right) ]\\
&=\left(4a-5b\right)\left(16a^{2}+20ab+25b^{2}\right)\\
&\mathbf{6. 729a^{3}-64b^{3}c^{6}}\\
&সমাধানঃ\\
&729a^{3}-64b^{3}c^{6}\\
&={\left(9a\right)}^{3}-{\left(4bc^{2}\right)}^{3}\\
&= \left(9a-4bc^{2}\right)\left\{{\left(9a\right)}^{2}+9a.4bc^{2}+{\left(4bc^{2}\right)}^{2}\right\} [ সূত্র a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+\ ab+b^{2}\right) ]\\
&=\left(9a-4bc^{2}\right)\left(81a^{2}+36abc^{2}+16b^{2}c^{4}\right)
\end{align*}
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\begin{align*}
&\mathbf{7. 27a^{3}b^{3}+64b^{3}c^{3}}\\
&সমাধানঃ\\
&27a^{3}b^{3}+64b^{3}c^{3}\\
&=b^{3}\left(27a^{3}+64c^{3}\right) [কমন]\\
&=b^{3}\left\{{\left(3a\right)}^{3}+{\left(4c\right)}^{3}\right\}\\
&= b^{3}\left(3a+4c\right)\left\{{\left(3a\right)}^{2}-3a.4c+{\left(4c\right)}^{2}\right\} [ সূত্র a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right) ]\\
&=b^{3}\left(3a+4c\right)\left(9a^{2}-12ac+16c^{2}\right)\\
&\mathbf{8. 56x^{3}-189y^{3}}\\
&সমাধানঃ\\
&56x^{3}-189y^{3}\\
&=7\left(8x^{3}-27y^{3}\right) [কমন]\\
&=7\left\{{\left(2x\right)}^{3}-{\left(3y\right)}^{3}\right\}\\
&= 7\left(2x-3y\right)\left\{{\left(2x\right)}^{2}+2x.3y+{\left(3y\right)}^{2}\right\} [ সূত্র a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+\ ab+b^{2}\right) ]\\
&=7\left(2x-3y\right)\left(4x^{2}+6xy+9y^{2}\right)\\
&\mathbf{9. 3x-75x^{3}}\\
&সমাধানঃ\\
&3x-75x^{3}\\
&=3x\left(1-25x^{2}\right) [কমন]\\
&=3x\left\{1^{2}-{\left(5x\right)}^{2}\right\}\\
&=3x\left(1+5x\right)\left(1-5x\right) \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right] \\
&\mathbf{10. 4x^{2}-y^{2}}\\
&সমাধানঃ\\
&4x^{2}-y^{2}\\
&={\left(2x\right)}^{2}-y^{2}\\
&=\left(2x+y\right)\left(2x-y\right) \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\
&\mathbf{11. 3ay^{2}-48a}\\
&সমাধানঃ\\
&3ay^{2}-48a\\
&=3a\left(y^{2}-16\right) [কমন]\\
&=3a\left(y^{2}-4^{2}\right)\\
&=3a\left(y+4\right)\left(y-4\right) \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right] \\
&\mathbf{12. a^{2}-2ab+b^{2}-p^{2}}\\
&সমাধানঃ\\
&a^{2}-2ab+b^{2}-p^{2}\\
&={\left(a-b\right)}^{2}-p^{2} [সূত্র\ {\left(a-b\right)}^{2}=a^{2}-2ab+b^{2}]\\
&=\left(a-b+p\right)\left(a-b-p\right) \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right] \\
&\mathbf{13. 16y^{2}-a^{2}-6a-9}\\
&সমাধানঃ\\
&16y^{2}-a^{2}-6a-9\\
&=16y^{2}-\left(a^{2}+6a+9\right)\\
&= 16y^{2}-\left(a^{2}+2.a.3+3^{2}\right)\\
&={\left(4y\right)}^{2}-{\left(a+3\right)}^{2} \left[সূত্র\ {\left(a+b\right)}^{2}=a^{2}+2ab+b^{2}\right] \\
&= \left\{4y+\left(a+3\right)\right\}\left\{4y-\left(a+3\right)\right\} \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right] \\
&=\left(4y+a+3\right)\left(4y-a-3\right)\\
&\mathbf{14. 8a+ap^{3}}\\
&সমাধানঃ\\
&8a+ap^{3}\\
&=a\left(8{+p}^{3}\right) [কমন]\\
&=a\left(p^{3}+8\right) \\
&=a\left\{p^{3}+2^{3}\right\}\\
&= a\left(p+2\right)\left(p^{2}-p.2+2^{2}\right) [ সূত্র a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right) ]\\
&=a\left(p+2\right)\left(p^{2}-2p+4\right)\\
&\mathbf{15. 2a^{3}+16b^{3}}\\
&সমাধানঃ\\
&2a^{3}+16b^{3}\\
&=2\left(a^{3}+8b^{3}\right) [কমন]\\
&=2\left\{a^{3}+{\left(2b\right)}^{3}\right\}\\
&= 2\left(a+2b\right)\left\{a^{2}-a.2b+{\left(2b\right)}^{2}\right\} [ সূত্র a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right) ]\\
&=2\left(a+2b\right)\left(a^{2}-2ab+4b^{2}\right)\\
&\mathbf{16. x^{2}+y^{2}-2xy-1}\\
&সমাধানঃ\\
&x^{2}+y^{2}-2xy-1\\
&= x^{2}-2xy+y^{2}-1\\
&={\left(x-y\right)}^{2}-1^{2} [সূত্র\ {\left(a-b\right)}^{2}=a^{2}-2ab+b^{2}]\\
&=\left(x-y+1\right)\left(x-y-1\right) \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\
&\mathbf{17. a^{2}-2ab+2b-1}\\
&সমাধানঃ\\
&a^{2}-2ab+2b-1\\
&= a^{2}-1-2ab+2b\\
&= a^{2}-1^{2}-2b\left(a-1\right) [কমন]\\
&= \left(a+1\right)\left(a-1\right)-2b\left(a-1\right) \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\
&= \left(a-1\right)\left(a+1-2b\right) [কমন]\\
&=\left(a-1\right)\left(a-2b+1\right)\\
&\mathbf{18. x^{4}-2x^{2}+1}\\
&সমাধানঃ\\
&x^{4}-2x^{2}+1\\
&= {\left(x^{2}\right)}^{2}-2.x^{2}.1+1^{2}\\
&={\left(x^{2}-1\right)}^{2} [সূত্র\ {\left(a-b\right)}^{2}=a^{2}-2ab+b^{2}]\\
&= {\left(x^{2}-1^{2}\right)}^{2}\\
&={\left\{\left(x+1\right)\left(x-1\right)\right\}}^{2} \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\
&={\left(x+1\right)}^{2}{\left(x-1\right)}^{2}\\
&\mathbf{19. 36-12x+x^{2}}\\
&সমাধানঃ\\
&36-12x+x^{2}\\
&=x^{2}-12x+36\\
&= x^{2}-2.x.6+6^{2}\\
&={\left(x-6\right)}^{2} [সূত্র\ {\left(a-b\right)}^{2}=a^{2}-2ab+b^{2}]\\
&\mathbf{20. x^{6}-y^{6}}\\
&সমাধানঃ\\
&x^{6}-y^{6}\\
&= {\left(x^{3}\right)}^{2}-{\left(y^{3}\right)}^{2}\\
&=\left(x^{3}+y^{3}\right)\left(x^{3}-y^{3}\right) \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\
&=\left(x+y\right)\left(x^{2}-xy+y^{2}\right)\left(x-y\right)\left(x^{2}+xy+y^{2}\right) \\
&[ সূত্র a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right) ] এবং [ সূত্র a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+\ ab+b^{2}\right) ]\\
&= \left(x+y\right)\left(x-y\right)\left(x^{2}-xy+y^{2}\right)\left(x^{2}+xy+y^{2}\right)\\
&\mathbf{21. {\left(x-y\right)}^{3}+z^{3}}\\
&সমাধানঃ\\
&{\left(x-y\right)}^{3}+z^{3}\\
&=\left\{\left(x-y\right)+z\right\}\left\{{\left(x-y\right)}^{2}-z\left(x-y\right)+z^{2}\right\} [ সূত্র a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right) ]\\
&=\left(x-y+z\right)\left(x^{2}-2xy+y^{2}-xz+yz+z^{2}\right) [সূত্র\ {\left(a-b\right)}^{2}=a^{2}-2ab+b^{2}]\\
&=\left(x-y+z\right)\left(x^{2}-2xy-xz+y^{2}+yz+z^{2}\right)\\
&\mathbf{22. 64x^{3}-8y^{3}}\\
&সমাধানঃ\\
&64x^{3}-8y^{3}\\
&=8\left(8x^{3}-y^{3}\right) [কমন]\\
&=8\left\{{\left(2x\right)}^{3}-y^{3}\right\}\\
&= 8\left(2x-y\right)\left\{{\left(2x\right)}^{2}+2x.y+y^{2}\right\} [ সূত্র a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+\ ab+b^{2}\right) ]\\
&=8\left(2x-y\right)\left(4x^{2}+2xy+y^{2}\right)\\
&\mathbf{23. x^{2}+14x+40}\\
&সমাধানঃ\\
&x^{2}+14x+40\\
&=x^{2}+10x+4x+40 [ মিডল টার্ম ব্রেক ]\\
&=x\left(x+10\right)+4\left(x+10\right) [কমন]\\
&=\left(x+10\right)\left(x+4\right) [কমন]\\
&\mathbf{24. x^{2}+7x-120}\\
&সমাধানঃ\\
&x^{2}+7x-120\\
&=x^{2}+15x-8x-120 [ মিডল টার্ম ব্রেক ]\\
&=x\left(x+15\right)-8\left(x+15\right) [কমন]\\
&=\left(x+15\right)\left(x-8\right) [কমন]\\
&\mathbf{25.x^{2}-51x+650}\\
&সমাধানঃ\\
&x^{2}-51x+650\\
&=x^{2}-26x-25x+650 [ মিডল টার্ম ব্রেক ]\\
&=x\left(x-26\right)-25\left(x-26\right) [কমন]\\
&=\left(x-26\right)\left(x-25\right) [কমন]\\
&\mathbf{26. a^{2}+7ab+12b^{2}}\\
&সমাধানঃ\\
&a^{2}+7ab+12b^{2}\\
&=a^{2}+4ab+3ab+12b^{2} [ মিডল টার্ম ব্রেক ]\\
&=a\left(a+4b\right)+3b\left(a+4b\right) [কমন]\\
&=\left(a+4b\right)\left(a+3b\right) [কমন]\\
&\mathbf{27. p^{2}+2pq-80q^{2}}\\
&সমাধানঃ\\
&p^{2}+2pq-80q^{2}\\
&=p^{2}+10pq-8pq-80q^{2} [ মিডল টার্ম ব্রেক ]\\
&=p\left(p+10q\right)-8q\left(p+10q\right) [কমন]\\
&=\left(p+10q\right)\left(p-8q\right) [কমন]\\
&\mathbf{28. x^{2}-3xy-40y^{2}}\\
&সমাধানঃ\\
&x^{2}-3xy-40y^{2}\\
&=x^{2}-8xy+5xy-40y^{2} [ মিডল টার্ম ব্রেক ]\\
&=x\left(x-8y\right)+5y\left(x-8y\right) [কমন]\\
&=\left(x-8y\right)\left(x+5y\right) [কমন]\\
&\mathbf{29. {\left(x^{2}-x\right)}^{2}+3\left(x^{2}-x\right)-40}\\
&সমাধানঃ\\
&ধরি, \quad t=x^{2}-x\\
&তাহলে, \quad t^{2}+3t-40\\
&=t^{2}+8t-5t-40 [ মিডল টার্ম ব্রেক ]\\
&=t\left(t+8\right)-5\left(t+8\right) [কমন]\\
&=\left(t+8\right)\left(t-5\right) [কমন]\\
&=\left(x^{2}-x+8\right)\left(x^{2}-x-5\right) [ t=x^{2}-x বসিয়ে] \\
&\mathbf{30. {\left(a^{2}+b^{2}\right)}^{2}-18\left(a^{2}+b^{2}\right)-88}\\
&সমাধানঃ\\
&ধরি, \quad t=a^{2}+b^{2}\\
&তাহলে, \quad t^{2}-18t-88\\
&=t^{2}-22t+4t-88 [ মিডল টার্ম ব্রেক ]\\
&=t\left(t-22\right)+4\left(t-22\right) [কমন]\\
&=\left(t-22\right)\left(t+4\right) [কমন]\\
&=\left(a^{2}+b^{2}-22\right)\left(a^{2}+b^{2}+4\right) [ t=a^{2}+b^{2} বসিয়ে]\\
&\mathbf{31. {\left(a^{2}+7a\right)}^{2}-8\left(a^{2}+7a\right)-180}\\
&সমাধানঃ\\
&ধরি, \quad t=a^{2}+7a\\
&তাহলে, \quad t^{2}-8t-180\\
&=t^{2}-18t+10t-180 [ মিডল টার্ম ব্রেক ]\\
&=t\left(t-18\right)+10\left(t-18\right) [কমন]\\
&=\left(t-18\right)\left(t+10\right) [কমন]\\
&=\left(a^{2}+7a-18\right)\left(a^{2}+7a+10\right) [ t=a^{2}+7a বসিয়ে]\\
&= \left(a^{2}+9a-2a-18\right)\left(a^{2}+5a+2a+10\right) [ মিডল টার্ম ব্রেক ]\\
&= \left\{a\left(a+9\right)-2\left(a+9\right)\right\}\left\{a\left(a+5\right)+2\left(a+5\right)\right\} [কমন]\\
&= \left(a+9\right)\left(a-2\right)\left(a+5\right)\left(a+2\right) [কমন]\\
&=\left(a+2\right)\left(a-2\right)\left(a+9\right)\left(a+5\right)\\
&\mathbf{32. x^{2}+\left(3a+4b\right)x+\left(2a^{2}+5ab+3b^{2}\right)}\\
&সমাধানঃ\\
&x^{2}+\left(3a+4b\right)x+\left(2a^{2}+5ab+3b^{2}\right)\\
&= x^{2}+\left(3a+4b\right)x+\left(2a^{2}+3ab+2ab+3b^{2}\right) [ মিডল টার্ম ব্রেক ]\\
&= x^{2}+\left(3a+4b\right)x+\left\{a\left(2a+3b\right)+b\left(2a+3b\right)\right\} [কমন]\\
&= x^{2}+\left(3a+4b\right)x+\left(2a+3b\right)\left(a+b\right) [কমন]\\
&=x^{2}+\left(2a+3b\right)x+\left(a+b\right)x+\left(a+b\right)\left(2a+3b\right) [ মিডল টার্ম ব্রেক ]\\
&=x\left(x+2a+3b\right)+\left(a+b\right)\left(x+2a+3b\right) [কমন]\\
&=\left(x+2a+3b\right)\left(x+a+b\right) [কমন]\\
&\mathbf{33. 6x^{2}-x-15}\\
&সমাধানঃ\\
&6x^{2}-x-15\\
&=6x^{2}-10x+9x-15 [ মিডল টার্ম ব্রেক ]\\
&=2x\left(3x-5\right)+3\left(3x-5\right) [কমন]\\
&=\left(3x-5\right)\left(2x+3\right) [কমন]\\
&\mathbf{34. x^{2}-x-\left(a+1\right)\left(a+2\right)}\\
&সমাধানঃ\\
&x^{2}-x-\left(a+1\right)\left(a+2\right)\\
&= x^{2}-\left(a+2\right)x+\left(a+1\right)x-\left(a+1\right)\left(a+2\right) [ মিডল টার্ম ব্রেক ]\\
&= x\left\{x-\left(a+2\right)\right\}+\left(a+1\right)\left\{x-\left(a+2\right)\right\} [কমন]\\
&= \left\{x-\left(a+2\right)\right\}\left\{x+\left(a+1\right)\right\} [কমন]\\
&= \left(x-a-2\right)\left(x+a+1\right)\\
&\mathbf{35. 3x^{2}+11x-4}\\
&সমাধানঃ\\
&3x^{2}+11x-4\\
&=3x^{2}+12x-x-4 [ মিডল টার্ম ব্রেক ]\\
&=3x\left(x+4\right)-1\left(x+4\right) [কমন]\\
&=\left(x+4\right)\left(3x-1\right) [কমন]\\
&\mathbf{36. 3x^{2}-16x-12}\\
&সমাধানঃ\\
&3x^{2}-16x-12\\
&=3x^{2}-18x+2x-12 [ মিডল টার্ম ব্রেক ]\\
&=3x\left(x-6\right)+2\left(x-6\right) [কমন]\\
&=\left(x-6\right)\left(3x+2\right) [কমন]\\
&\mathbf{37. 2x^{2}-9x-35}\\
&সমাধানঃ\\
&2x^{2}-9x-35\\
&=2x^{2}-14x+5x-35 [ মিডল টার্ম ব্রেক ]\\
&=2x\left(x-7\right)+5\left(x-7\right) [কমন]\\
&=\left(x-7\right)\left(2x+5\right) [কমন]\\
&\mathbf{38. 2x^{2}-5xy+2y^{2}}\\
&সমাধানঃ\\
&2x^{2}-5xy+2y^{2}\\
&=2x^{2}-4xy-xy+2y^{2} [ মিডল টার্ম ব্রেক ]\\
&=2x\left(x-2y\right)-y\left(x-2y\right) [কমন]\\
&=\left(x-2y\right)\left(2x-y\right) [কমন]\\
&\mathbf{39. x^{3}-8{\left(x-y\right)}^{3}}\\
&সমাধানঃ\\
&x^{3}-8{\left(x-y\right)}^{3}\\
&=x^{3}-{\left\{2\left(x-y\right)\right\}}^{3}\\
&= \left\{x-2\left(x-y\right)\right\}\left[x^{2}+x.2\left(x-y\right)+{\left\{2\left(x-y\right)\right\}}^{2}\right] [ সূত্র a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+\ ab+b^{2}\right)]\\
&=\left(x-2x+2y\right)\left\{x^{2}+2x\left(x-y\right)+4{\left(x-y\right)}^{2}\right\}\\
&= \left(-x+2y\right)\left\{x^{2}+2x^{2}-2xy+4\left(x^{2}-2xy+y^{2}\right)\right\} [সূত্র\ {\left(a-b\right)}^{2}=a^{2}-2ab+b^{2}]\\
&= \left(2y-x\right)\left(x^{2}+2x^{2}-2xy+4x^{2}-8xy+4y^{2}\right)\\
&= \left(2y-x\right)\left(7x^{2}-10xy+4y^{2}\right)\\
&\mathbf{40. 10p^{2}+11pq-6q^{2}}\\
&সমাধানঃ\\
&10p^{2}+11pq-6q^{2}\\
&=10p^{2}+15pq-4pq-6q^{2} [ মিডল টার্ম ব্রেক ]\\
&=5p\left(2p+3q\right)-2q\left(2p+3q\right) [কমন]\\
&=\left(2p+3q\right)\left(5p-2q\right) [কমন]\\
&\mathbf{41. 2{\left(x+y\right)}^{2}-3\left(x+y\right)-2}\\
&সমাধানঃ\\
&ধরি, t=x+y\\
&তাহলে, \quad 2t^{2}-3t-2\\
&=2t^{2}-4t+t-2 [ মিডল টার্ম ব্রেক ]\\
&=2t\left(t-2\right)+\left(t-2\right) [কমন]\\
&=\left(2t+1\right)\left(t-2\right) [কমন]\\
&= \left\{2\left(x+y\right)+1\right\}\left(x+y-2\right) [ t=x+y বসিয়ে]\\
&=\left(2x+2y+1\right)\left(x+y-2\right)\\
&\mathbf{42. ax^{2}+\left(a^{2}+1\right)x+a}\\
&সমাধানঃ\\
&ax^{2}+\left(a^{2}+1\right)x+a\\
&=ax^{2}+a^{2}x+x+a\\
&=ax\left(x+a\right)+1\left(x+a\right) [কমন]\\
&=\left(x+a\right)\left(ax+1\right) [কমন]\\
&\mathbf{43. 15x^{2}-11xy-12y^{2}}\\
&সমাধানঃ\\
&15x^{2}-11xy-12y^{2}\\
&=15x^{2}-20xy+9xy-12y^{2} [ মিডল টার্ম ব্রেক ]\\
&=5x\left(3x-4y\right)+3y\left(3x-4y\right) [কমন]\\
&=\left(3x-4y\right)\left(5x+3y\right) [কমন]\\
&\mathbf{44. a^{3}-3a^{2}b+3ab^{2}-2b^{3}}\\
&সমাধানঃ\\
&a^{3}-3a^{2}b+3ab^{2}-2b^{3}\\
&= a^{3}-3a^{2}b+3ab^{2}-b^{3}-b^{3}\\
&={\left(a-b\right)}^{3}-b^{3} [ সূত্র {\left(a-b\right)}^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} ]\\
&=\left(a-b-b\right)\left\{{\left(a-b\right)}^{2}+\left(a-b\right).b+b^{2}\right\}\\
&= \left(a-2b\right)\left(a^{2}-2ab+b^{2}+ab-b^{2}+b^{2}\right) [সূত্র\ {\left(a-b\right)}^{2}=a^{2}-2ab+b^{2}]\\
&=\left(a-2b\right)\left(a^{2}-ab+b^{2}\right)
\end{align*}