Koshe Dekhi 8.4 Class 9

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

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

সমাধানঃ

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

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

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

সমাধানঃ

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

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

সমাধানঃ

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

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

সমাধানঃ

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

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

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

সমাধানঃ

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

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

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

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

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

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

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

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

$=0\times$

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

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

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

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

সমাধানঃ

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

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

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

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

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

$-\left ( 2y-x \right )\left ( 2x-y \right )$ ]

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

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

$=0\times$

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

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

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

$-\left ( 2y-x \right )\left ( 2x-y \right )$ ]

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

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

সমাধানঃ

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

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

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

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

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

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

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

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

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

সমাধানঃ

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

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

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

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

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

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

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

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

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

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

সমাধানঃ

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

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

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

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

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

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

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

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

$=0\times$

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

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

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

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

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

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

উৎপাদকে বিশ্লেষণ : কষে দেখি - 8.4

নিচের বহুপদী সংখ্যামালাগুলিকে উৎপাদকে বিশ্লেষণ করি :

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

সমাধানঃ

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

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

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

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

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

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

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

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

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

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

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

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

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

Koshe dekhi 8.4 Class 9

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