# Koshe Dekhi 8.4Class 9

### Koshe Dekhi 8.4 Class 9

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

1. $\fn_cm&space;{\color{Blue}&space;x^{3}+y^{3}-12xy+64}$

সমাধানঃ

$\fn_cm&space;x^{3}+y^{3}-12xy+64$

$\fn_cm&space;=x^{3}+y^{3}+64-12xy$

$=\left&space;(&space;x&space;\right&space;)^{3}+\left&space;(&space;y&space;\right&space;)^{3}+\left&space;(&space;4&space;\right&space;)^{3}-3\times&space;x\times&space;y\times&space;4$

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

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

$=\left&space;(&space;x+y+4&space;\right&space;)\left&space;(&space;x^{2}+&space;y^{2}+16-xy-4y-4x&space;\right&space;)$  (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

2. $\fn_cm&space;{\color{Blue}&space;8x^{3}-y^{3}+1+6xy}$

সমাধানঃ

$\fn_cm&space;8x^{3}-y^{3}+1+6xy$

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

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

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

$=\left&space;(&space;2x-y+1&space;\right&space;)\left&space;(&space;4x^{2}+&space;y^{2}+1+2xy+y-2x&space;\right&space;)$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

3. $\fn_cm&space;{\color{Blue}&space;8a^{3}-27b^{3}-1-18ab}$

সমাধানঃ

$\fn_cm&space;8a^{3}-27b^{3}-1-18ab$

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

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

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

$=\left&space;(&space;2a-3b-1&space;\right&space;)\left&space;(&space;4a^{2}+&space;9b^{2}+1+6ab-3b+2a&space;\right&space;)$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

4. $\fn_cm&space;{\color{Blue}&space;1+8x^{3}+18xy-27y^{3}}$

সমাধানঃ

$\fn_cm&space;1+8x^{3}+18xy-27y^{3}$

$=8x^{3}-27y^{3}+1+18xy$

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

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

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

$=\left&space;(&space;2x-3y+1&space;\right&space;)\left&space;(&space;4x^{2}+&space;9y^{2}+1+6xy+3y-2x&space;\right&space;)$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

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

সমাধানঃ

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

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

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

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

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

$-\left&space;(&space;5c-3a&space;\right&space;)\times&space;\left&space;(3a-2b&space;\right&space;)$ ]

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

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

$=0\times$

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

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

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

$-\left&space;(&space;5c-3a&space;\right&space;)\times&space;\left&space;(3a-2b&space;\right&space;)$ ]

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

$=0+3\times&space;\left&space;(&space;3a-2b&space;\right&space;)\times&space;\left&space;(&space;2b-5c&space;\right&space;)\times&space;\left&space;(&space;5c-3a&space;\right&space;)$

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

6. $\fn_cm&space;{\color{Blue}&space;\left&space;(&space;2x-y&space;\right&space;)^{3}-\left&space;(&space;x+y&space;\right&space;)^{3}+\left&space;(&space;2y-x&space;\right&space;)^{3}}$

সমাধানঃ

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

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

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

$-\left&space;(&space;2x-y&space;\right&space;)\left&space;\{&space;-\left&space;(&space;x+y&space;\right&space;)&space;\right&space;\}$

$-\left&space;\{&space;-\left&space;(&space;x+y&space;\right&space;)&space;\right&space;\}\left&space;(&space;2y-x&space;\right&space;)$

$-\left&space;(&space;2y-x&space;\right&space;)\left&space;(&space;2x-y&space;\right&space;)$ ]

$+3\times&space;\left&space;(&space;2x-y&space;\right&space;)\times&space;\left&space;\{&space;-\left&space;(&space;x+y&space;\right&space;)&space;\right&space;\}\times&space;\left&space;(&space;2y-x&space;\right&space;)$

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

$=0\times$

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

$-\left&space;(&space;2x-y&space;\right&space;)\left&space;\{&space;-\left&space;(&space;x+y&space;\right&space;)&space;\right&space;\}$

$-\left&space;\{&space;-\left&space;(&space;x+y&space;\right&space;)&space;\right&space;\}\left&space;(&space;2y-x&space;\right&space;)$

$-\left&space;(&space;2y-x&space;\right&space;)\left&space;(&space;2x-y&space;\right&space;)$ ]

$+3\times&space;\left&space;(&space;2x-y&space;\right&space;)\times&space;\left&space;\{&space;-\left&space;(&space;x+y&space;\right&space;)&space;\right&space;\}\times&space;\left&space;(&space;2y-x&space;\right&space;)$

$=0+3\times&space;\left&space;(&space;2x-y&space;\right&space;)\times&space;\left&space;\{&space;-\left&space;(&space;x+y&space;\right&space;)&space;\right&space;\}\times&space;\left&space;(&space;2y-x&space;\right&space;)$

$=-3\left&space;(&space;2x-y&space;\right&space;)\left&space;(&space;x+y&space;\right&space;)&space;\left&space;(&space;2y-x&space;\right&space;)$

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

$=3\left&space;(&space;2x-y&space;\right&space;)\left&space;(&space;x+y&space;\right&space;)&space;\left&space;(&space;x-2y&space;\right&space;)$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

7. $\fn_cm&space;{\color{Blue}&space;a^{6}+32a^{3}-64}$

সমাধানঃ

$a^{6}+32a^{3}-64$

$=a^{6}+8a^{3}+24a^{3}-64$

$=a^{6}+8a^{3}-64+24a^{3}$

$=\left&space;(&space;a^{2}&space;\right&space;)^{3}+\left&space;(&space;2a&space;\right&space;)^{3}+\left&space;(&space;-4&space;\right&space;)^{3}+24a^{3}$

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

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

$+3\times&space;a^{2}\times&space;2a\times&space;\left&space;(&space;-4&space;\right&space;)+24a^{3}$

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

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

$\left&space;(&space;a^{4}+4a^{2}+16-2a^{3}+8a+4a^{2}\right&space;)$

$-24a^{3}+24a^{3}$

$=\left&space;(&space;a^{2}+2a-4&space;\right&space;)\left&space;(&space;a^{4}-2a^{3}+8a^{2}+8a+16\right&space;)$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

8. $\fn_cm&space;{\color{Blue}&space;a^{6}-18a^{3}+125}$

সমাধানঃ

$a^{6}-18a^{3}+125$

$=a^{6}+27a^{3}-27a^{3}-18a^{3}+125$

$=a^{6}+27a^{3}+125-27a^{3}-18a^{3}$

$=\left&space;[a^{6}+27a^{3}+125&space;\right&space;]-45a^{3}$

$=\left&space;[\left&space;(&space;a^{2}&space;\right&space;)^{3}+\left&space;(&space;3a&space;\right&space;)^{3}+\left&space;(&space;5&space;\right&space;)^{3}&space;\right&space;]-45a^{3}$

$=\left&space;(&space;a^{2}+3a+5&space;\right&space;)$

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

$+3\times&space;a^{2}\times&space;3a\times&space;5-45a^{3}$

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

$=\left&space;(&space;a^{2}+3a+5&space;\right&space;)$

$\left&space;[&space;a^{4}&space;+9a^{2}+25-3a^{3}-15a-5a^{2}&space;\right&space;]$

$+45a^{3}-45a^{3}$

$=\left&space;(&space;a^{2}+3a+5&space;\right&space;)\left&space;(&space;a^{4}-3a^{3}&space;+4a^{2}-15a+25\right&space;)$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

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

সমাধানঃ

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

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

$\fn_cm&space;=\left&space;(&space;pq-pr&space;\right&space;)^{3}+\left&space;(&space;qr-pq&space;\right&space;)^{3}+\left&space;(&space;pr-qr&space;\right&space;)^{3}$

$=\left&space;(&space;pq-pr+qr-pq+pr-qr&space;\right&space;)$

[$\left&space;(&space;pq-pr&space;\right&space;)^{2}+\left&space;(&space;qr-pq&space;\right&space;)^{2}+\left&space;(&space;pr-qr&space;\right&space;)^{2}$

$-\left&space;(&space;pq-pr&space;\right&space;)\times&space;\left&space;(&space;qr-pq&space;\right&space;)-\left&space;(&space;qr-pq&space;\right&space;)\times&space;\left&space;(&space;pr-qr&space;\right&space;)-\left&space;(&space;pr-qr&space;\right&space;)\times&space;\left&space;(&space;pq-pr&space;\right&space;)$]

$+3\times&space;\left&space;(&space;pq-pr&space;\right&space;)\times&space;\left&space;(&space;qr-pq&space;\right&space;)\times&space;\left&space;(&space;pr-qr&space;\right&space;)$

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

$=0\times$

[$\left&space;(&space;pq-pr&space;\right&space;)^{2}+\left&space;(&space;qr-pq&space;\right&space;)^{2}+\left&space;(&space;pr-qr&space;\right&space;)^{2}$

$-\left&space;(&space;pq-pr&space;\right&space;)\times&space;\left&space;(&space;qr-pq&space;\right&space;)-\left&space;(&space;qr-pq&space;\right&space;)\times&space;\left&space;(&space;pr-qr&space;\right&space;)-\left&space;(&space;pr-qr&space;\right&space;)\times&space;\left&space;(&space;pq-pr&space;\right&space;)$]

$+3\times&space;\left&space;(&space;pq-pr&space;\right&space;)\times&space;\left&space;(&space;qr-pq&space;\right&space;)\times&space;\left&space;(&space;pr-qr&space;\right&space;)$

$=0+3\times&space;\left&space;(&space;pq-pr&space;\right&space;)\times&space;\left&space;(&space;qr-pq&space;\right&space;)\times&space;\left&space;(&space;pr-qr&space;\right&space;)$

$\fn_cm&space;=3\times&space;\left&space;[p\left&space;(&space;q-r&space;\right&space;)&space;\right&space;]\times&space;\left&space;[q\left&space;(&space;r-p&space;\right&space;)&space;\right&space;]\times&space;\left&space;[r\left&space;(&space;p-q&space;\right&space;)&space;\right&space;]$

$\fn_cm&space;=3pqr&space;\left&space;(&space;q-r&space;\right&space;)\left&space;(&space;r-p&space;\right&space;)\left&space;(&space;p-q&space;\right&space;)$ (উত্তর)

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

10. $\fn_cm&space;{\color{Blue}&space;p^{3}+\frac{1}{p^{3}}+\frac{26}{27}}$

সমাধানঃ

$p^{3}+\frac{1}{p^{3}}+\frac{26}{27}$

$=p^{3}+\frac{1}{p^{3}}-\frac{1}{27}+\frac{1}{27}+\frac{26}{27}$

$=\left&space;[\left&space;(&space;p&space;\right&space;)^{3}+\left&space;(&space;\frac{1}{p}&space;\right&space;)^{3}+\left&space;(&space;-\frac{1}{3}&space;\right&space;)^{3}&space;\right&space;]+\frac{1}{27}+\frac{26}{27}$

$=\left&space;(&space;p+\frac{1}{p}-\frac{1}{3}&space;\right&space;)$

$\left&space;[&space;\left&space;(&space;p&space;\right&space;)^{2}+\left&space;(&space;\frac{1}{p}&space;\right&space;)^{2}+\left&space;(&space;-\frac{1}{3}&space;\right&space;)^{2}-p\times&space;\frac{1}{p}-\frac{1}{p}\times&space;\left&space;(&space;-\frac{1}{3}&space;\right&space;)-\left&space;(&space;-\frac{1}{3}&space;\right&space;)\times&space;p&space;\right&space;]$

$+3\times&space;p\times&space;\frac{1}{p}\times&space;\left&space;(&space;-\frac{1}{3}&space;\right&space;)+\frac{1+26}{27}$

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

$=\left&space;(&space;p+\frac{1}{p}-\frac{1}{3}&space;\right&space;)$

$\left&space;(&space;p^{2}+\frac{1}{p^{2}}+\frac{1}{9}-1+\frac{1}{3p}+\frac{p}{3}&space;\right&space;)$

$-1+\frac{27}{27}$

$=\left&space;(&space;p+\frac{1}{p}-\frac{1}{3}&space;\right&space;)$

$\left&space;(&space;p^{2}+\frac{1}{p^{2}}+\frac{1-9}{9}+\frac{1}{3p}+\frac{p}{3}&space;\right&space;)-1+1$

$=\left&space;(&space;p+\frac{1}{p}-\frac{1}{3}&space;\right&space;)\left&space;(&space;p^{2}+\frac{1}{p^{2}}-\frac{8}{9}+\frac{1}{3p}+\frac{p}{3}&space;\right&space;)$ (উত্তর)

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