অষ্টম শ্রেণি – চতুর্থ অধ্যায় : বহুপদী সংখ্যামালার গুণ ও ভাগ সম্পূর্ণ সমাধান

কষে দেখি – 4.1

1.

প্রথম বীজগাণিতিক সংখ্যামালা	দ্বিতীয় বীজগাণিতিক সংখ্যামালা	গুণফল	গুণফলের মান
\[\left( a \right){{x}^{2}}-3x+5\]	\[5x+9\]	\[5{{x}^{3}}-6{{x}^{2}}-2x+45\]	x = 1 বসিয়ে পেলাম 42
\[\left( b \right){{x}^{2}}+12-7y\]	\[2x-y\]		x = -2 ও y = 2 বসিয়ে পেলাম _____
\[\left( c \right)8{{p}^{3}}-3p-2{{p}^{2}}\]	\[4{{p}^{2}}-5\]		p = -2  বসিয়ে পেলাম _____
\[\left( d \right)6a+5b+2\]	\[a-b+6\]		a = 0 ও b = -1  বসিয়ে পেলাম _____
\[\left( e \right){{p}^{3}}-{{p}^{2}}{{q}^{2}}+{{q}^{3}}\]	\[{{p}^{2}}+pq+{{q}^{2}}\]		p = 2 ও  q = -2  বসিয়ে পেলাম _____
\[\left( f \right){{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xy-yz-zx\]	\[x+y+z\]		x = 1, y = 0, z = -1  বসিয়ে পেলাম ______

উত্তরঃ-

প্রথম বীজগাণিতিক সংখ্যামালা	দ্বিতীয় বীজগাণিতিক সংখ্যামালা	গুণফল	গুণফলের মান
\[\left( a \right){{x}^{2}}-3x+5\]	\[5x+9\]	\[5{{x}^{3}}-6{{x}^{2}}-2x+45\]	x = 1 বসিয়ে পেলাম 42
\[\left( b \right){{x}^{2}}+12-7y\]	\[2x-y\]	\[2{{x}^{3}}-{{x}^{2}}y-14xy+24x-12y+7{{y}^{2}}\]	x = -2 ও y = 2 বসিয়ে পেলাম -12
\[\left( c \right)8{{p}^{3}}-3p-2{{p}^{2}}\]	\[4{{p}^{2}}-5\]	\[32{{p}^{5}}-8{{p}^{4}}-52{{p}^{3}}+10{{p}^{2}}+15p\]	p = -2  বসিয়ে পেলাম 726
\[\left( d \right)6a+5b+2\]	\[a-b+6\]	\[6{{a}^{2}}+38a-ab+28b-5{{b}^{2}}+12\]	a = 0 ও b = -1  বসিয়ে পেলাম -21
\[\left( e \right){{p}^{3}}-{{p}^{2}}{{q}^{2}}+{{q}^{3}}\]	\[{{p}^{2}}+pq+{{q}^{2}}\]	\[{{p}^{5}}+{{p}^{4}}q-{{p}^{4}}{{q}^{2}}-{{p}^{3}}{{q}^{3}}+{{p}^{3}}{{q}^{2}}-{{p}^{2}}{{q}^{4}}+{{p}^{2}}{{q}^{3}}+p{{q}^{4}}+{{q}^{5}}\]	p = 2 ও  q = -2  বসিয়ে পেলাম -64
\[\left( f \right){{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xy-yz-zx\]	\[x+y+z\]	\[{{x}^{3}}-3xyz+{{y}^{3}}+{{z}^{3}}\]	x = 1, y = 0, z = -1  বসিয়ে পেলাম 0

2. ধারাবাহিক গুণ করে গুণফল খুঁজি (পরপর গুণ করি)

\[\left( i \right)\left( {{x}^{5}}+1 \right)\times \left( 3-{{x}^{4}} \right)\times \left( 4+{{x}^{3}}+{{x}^{6}} \right)\]

\[\left( ii \right)\left( 2{{a}^{3}}-3{{b}^{5}} \right)\times \left( 2{{a}^{3}}+3{{b}^{5}} \right)\times \left( 2{{a}^{4}}-3{{a}^{2}}{{b}^{2}}+{{b}^{4}} \right)\]

\[(iii)\left( ax+by \right)\times \left( ax-by \right)\times \left( {{a}^{4}}{{x}^{4}}+{{a}^{2}}{{b}^{2}}{{x}^{2}}{{y}^{2}}+{{b}^{4}}{{y}^{4}} \right)\]

\[(iv)\left( a+b+c \right)\times \left( a-b+c \right)\times \left( a+b-c \right)\]

\[(v)\left( \frac{2{{p}^{2}}}{{{q}^{2}}}+\frac{5{{q}^{2}}}{{{p}^{2}}} \right)\left( \frac{2{{p}^{2}}}{{{q}^{2}}}-\frac{5{{q}^{2}}}{{{p}^{2}}} \right)\]

\[(vi)\left( \frac{{{x}^{2}}}{{{y}^{2}}}+\frac{{{y}^{2}}}{{{z}^{2}}} \right)\times \left( \frac{{{y}^{2}}}{{{z}^{2}}}+\frac{{{z}^{2}}}{{{x}^{2}}} \right)\times \left( \frac{{{z}^{2}}}{{{x}^{2}}}+\frac{{{x}^{2}}}{{{y}^{2}}} \right)\]

উত্তরঃ-

\[\left( i \right)\left( {{x}^{5}}+1 \right)\times \left( 3-{{x}^{4}} \right)\times \left( 4+{{x}^{3}}+{{x}^{6}} \right)\]

\[=\left\{ {{x}^{5}}\left( 3-{{x}^{4}} \right)+1\left( 3-{{x}^{4}} \right) \right\}\left( 4+{{x}^{3}}+{{x}^{6}} \right)\]

\[=\left( 3{{x}^{5}}-{{x}^{9}}+3-{{x}^{4}} \right)\left( 4+{{x}^{3}}+{{x}^{6}} \right)\]

\[=3{{x}^{5}}\left( 4+{{x}^{3}}+{{x}^{6}} \right)-{{x}^{9}}\left( 4+{{x}^{3}}+{{x}^{6}} \right)+3\left( 4+{{x}^{3}}+{{x}^{6}} \right)-{{x}^{4}}\left( 4+{{x}^{3}}+{{x}^{6}} \right)\]

\[=12{{x}^{5}}+3{{x}^{8}}+3{{x}^{11}}-4{{x}^{9}}-{{x}^{12}}-{{x}^{15}}+12+3{{x}^{3}}+3{{x}^{6}}-4{{x}^{4}}-{{x}^{7}}-{{x}^{10}}\]

\[=-{{x}^{15}}-{{x}^{12}}+3{{x}^{11}}-{{x}^{10}}-4{{x}^{9}}+3{{x}^{8}}-{{x}^{7}}+3{{x}^{6}}+12{{x}^{5}}-4{{x}^{4}}+3{{x}^{3}}+12\]

\[\left( ii \right)\left( 2{{a}^{3}}-3{{b}^{5}} \right)\times \left( 2{{a}^{3}}+3{{b}^{5}} \right)\times \left( 2{{a}^{4}}-3{{a}^{2}}{{b}^{2}}+{{b}^{4}} \right)\]

\[=\left\{ 2{{a}^{3}}\left( 2{{a}^{3}}+3{{b}^{5}} \right)-3{{b}^{5}}\left( 2{{a}^{3}}+3{{b}^{5}} \right) \right\}\left( 2{{a}^{4}}-3{{a}^{2}}{{b}^{2}}+{{b}^{4}} \right)\]

\[=\left( 4{{a}^{6}}+6{{a}^{3}}{{b}^{5}}-6{{a}^{3}}{{b}^{5}}-9{{b}^{10}} \right)\left( 2{{a}^{4}}-3{{a}^{2}}{{b}^{2}}+{{b}^{4}} \right)\]

\[=\left( 4{{a}^{6}}-9{{b}^{10}} \right)\left( 2{{a}^{4}}-3{{a}^{2}}{{b}^{2}}+{{b}^{4}} \right)\]

\[=4{{a}^{6}}\left( 2{{a}^{4}}-3{{a}^{2}}{{b}^{2}}+{{b}^{4}} \right)-9{{b}^{10}}\left( 2{{a}^{4}}-3{{a}^{2}}{{b}^{2}}+{{b}^{4}} \right)\]

\[=8{{a}^{10}}-12{{a}^{8}}{{b}^{2}}+4{{a}^{6}}{{b}^{4}}-18{{a}^{4}}{{b}^{10}}+27{{a}^{2}}{{b}^{12}}-9{{b}^{14}}\]

\[(iii)\left( ax+by \right)\times \left( ax-by \right)\times \left( {{a}^{4}}{{x}^{4}}+{{a}^{2}}{{b}^{2}}{{x}^{2}}{{y}^{2}}+{{b}^{4}}{{y}^{4}} \right)\]

\[=\left\{ ax\left( ax-by \right)+by\left( ax-by \right) \right\}\left( {{a}^{4}}{{x}^{4}}+{{a}^{2}}{{b}^{2}}{{x}^{2}}{{y}^{2}}+{{b}^{4}}{{y}^{4}} \right)\]

\[=\left\{ {{a}^{2}}{{x}^{2}}-axby+axby-{{b}^{2}}{{y}^{2}} \right\}\left( {{a}^{4}}{{x}^{4}}+{{a}^{2}}{{b}^{2}}{{x}^{2}}{{y}^{2}}+{{b}^{4}}{{y}^{4}} \right)\]

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

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

\[={{a}^{6}}{{x}^{6}}+{{a}^{4}}{{b}^{2}}{{x}^{4}}{{y}^{2}}+{{a}^{2}}{{b}^{4}}{{x}^{2}}{{y}^{4}}-{{a}^{4}}{{b}^{2}}{{x}^{4}}{{y}^{2}}-{{a}^{2}}{{b}^{4}}{{x}^{2}}{{y}^{4}}-{{b}^{6}}{{y}^{6}}\]

\[={{a}^{6}}{{x}^{6}}-{{b}^{6}}{{y}^{6}}\]

\[(iv)\left( a+b+c \right)\times \left( a-b+c \right)\times \left( a+b-c \right)\]

\[=\left\{ a\left( a-b+c \right)+b\left( a-b+c \right)+c\left( a-b+c \right) \right\}\left( a+b-c \right)\]

\[=\left\{ {{a}^{2}}-ab+ac+ab-{{b}^{2}}+bc+ac-bc+{{c}^{2}} \right\}\left( a+b-c \right)\]

\[=\left\{ {{a}^{2}}+2ac-{{b}^{2}}+{{c}^{2}} \right\}\left( a+b-c \right)\]

\[=\left\{ {{a}^{2}}+2ac-{{b}^{2}}+{{c}^{2}} \right\}a+\left\{ {{a}^{2}}+2ac-{{b}^{2}}+{{c}^{2}} \right\}b-\left\{ {{a}^{2}}+2ac-{{b}^{2}}+{{c}^{2}} \right\}c\]

\[={{a}^{3}}+2{{a}^{2}}c-a{{b}^{2}}+a{{c}^{2}}+{{a}^{2}}b+2abc-{{b}^{3}}+b{{c}^{2}}-{{a}^{2}}c-2a{{c}^{2}}+{{b}^{2}}c-{{c}^{3}}\]

\[={{a}^{3}}+{{a}^{2}}c-a{{b}^{2}}-a{{c}^{2}}+{{a}^{2}}b+2abc+b{{c}^{2}}+{{b}^{2}}c-{{b}^{3}}-{{c}^{3}}\]

\[(v)\left( \frac{2{{p}^{2}}}{{{q}^{2}}}+\frac{5{{q}^{2}}}{{{p}^{2}}} \right)\left( \frac{2{{p}^{2}}}{{{q}^{2}}}-\frac{5{{q}^{2}}}{{{p}^{2}}} \right)\]

\[=\frac{2{{p}^{2}}}{{{q}^{2}}}\left( \frac{2{{p}^{2}}}{{{q}^{2}}}-\frac{5{{q}^{2}}}{{{p}^{2}}} \right)+\frac{5{{q}^{2}}}{{{p}^{2}}}\left( \frac{2{{p}^{2}}}{{{q}^{2}}}-\frac{5{{q}^{2}}}{{{p}^{2}}} \right)\]

\[=\frac{4{{p}^{4}}}{{{q}^{4}}}-\frac{2{{p}^{2}}}{{{q}^{2}}}\times \frac{5{{q}^{2}}}{{{p}^{2}}}+\frac{2{{p}^{2}}}{{{q}^{2}}}\times \frac{5{{q}^{2}}}{{{p}^{2}}}-\frac{25{{q}^{4}}}{{{p}^{4}}}\]

\[=\frac{4{{p}^{4}}}{{{q}^{4}}}-\frac{25{{q}^{4}}}{{{p}^{4}}}\]

\[(vi)\left( \frac{{{x}^{2}}}{{{y}^{2}}}+\frac{{{y}^{2}}}{{{z}^{2}}} \right)\times \left( \frac{{{y}^{2}}}{{{z}^{2}}}+\frac{{{z}^{2}}}{{{x}^{2}}} \right)\times \left( \frac{{{z}^{2}}}{{{x}^{2}}}+\frac{{{x}^{2}}}{{{y}^{2}}} \right)\]

\[=\left\{ \frac{{{x}^{2}}}{{{y}^{2}}}\left( \frac{{{y}^{2}}}{{{z}^{2}}}+\frac{{{z}^{2}}}{{{x}^{2}}} \right)+\frac{{{y}^{2}}}{{{z}^{2}}}\left( \frac{{{y}^{2}}}{{{z}^{2}}}+\frac{{{z}^{2}}}{{{x}^{2}}} \right) \right\}\left( \frac{{{z}^{2}}}{{{x}^{2}}}+\frac{{{x}^{2}}}{{{y}^{2}}} \right)\]

\[=\left\{ \frac{{{x}^{2}}}{{{y}^{2}}}\times \frac{{{y}^{2}}}{{{z}^{2}}}+\frac{{{x}^{2}}}{{{y}^{2}}}\times \frac{{{z}^{2}}}{{{x}^{2}}}+\frac{{{y}^{4}}}{{{z}^{4}}}+\frac{{{y}^{2}}}{{{z}^{2}}}\times \frac{{{z}^{2}}}{{{x}^{2}}} \right\}\left( \frac{{{z}^{2}}}{{{x}^{2}}}+\frac{{{x}^{2}}}{{{y}^{2}}} \right)\]

\[=\left( \frac{{{x}^{2}}}{{{z}^{2}}}+\frac{{{z}^{2}}}{{{y}^{2}}}+\frac{{{y}^{4}}}{{{z}^{4}}}+\frac{{{y}^{2}}}{{{x}^{2}}} \right)\left( \frac{{{z}^{2}}}{{{x}^{2}}}+\frac{{{x}^{2}}}{{{y}^{2}}} \right)\]

\[=\left\{ \left( \frac{{{x}^{2}}}{{{z}^{2}}}+\frac{{{z}^{2}}}{{{y}^{2}}}+\frac{{{y}^{4}}}{{{z}^{4}}}+\frac{{{y}^{2}}}{{{x}^{2}}} \right)\frac{{{z}^{2}}}{{{x}^{2}}} \right\}+\left\{ \left( \frac{{{x}^{2}}}{{{z}^{2}}}+\frac{{{z}^{2}}}{{{y}^{2}}}+\frac{{{y}^{4}}}{{{z}^{4}}}+\frac{{{y}^{2}}}{{{x}^{2}}} \right)\frac{{{x}^{2}}}{{{y}^{2}}} \right\}\]

\[=\left\{ \frac{{{x}^{2}}}{{{z}^{2}}}\times \frac{{{z}^{2}}}{{{x}^{2}}}+\frac{{{z}^{2}}}{{{y}^{2}}}\times \frac{{{z}^{2}}}{{{x}^{2}}}+\frac{{{y}^{4}}}{{{z}^{4}}}\times \frac{{{z}^{2}}}{{{x}^{2}}}+\frac{{{y}^{2}}}{{{x}^{2}}}\times \frac{{{z}^{2}}}{{{x}^{2}}} \right\}+\left\{ \frac{{{x}^{2}}}{{{z}^{2}}}\times \frac{{{x}^{2}}}{{{y}^{2}}}+\frac{{{z}^{2}}}{{{y}^{2}}}\times \frac{{{x}^{2}}}{{{y}^{2}}}+\frac{{{y}^{4}}}{{{z}^{4}}}\times \frac{{{x}^{2}}}{{{y}^{2}}}+\frac{{{y}^{2}}}{{{x}^{2}}}\times \frac{{{x}^{2}}}{{{y}^{2}}} \right\}\]

\[=1+\frac{{{z}^{4}}}{{{x}^{2}}{{y}^{2}}}+\frac{{{y}^{4}}}{{{x}^{2}}{{z}^{2}}}+\frac{{{y}^{2}}{{z}^{2}}}{{{x}^{4}}}+\frac{{{x}^{4}}}{{{y}^{2}}{{z}^{2}}}+\frac{{{x}^{2}}{{z}^{2}}}{{{y}^{4}}}+\frac{{{x}^{2}}{{y}^{2}}}{{{z}^{4}}}+1\]

\[=2+\frac{{{z}^{4}}}{{{x}^{2}}{{y}^{2}}}+\frac{{{y}^{4}}}{{{x}^{2}}{{z}^{2}}}+\frac{{{y}^{2}}{{z}^{2}}}{{{x}^{4}}}+\frac{{{x}^{4}}}{{{y}^{2}}{{z}^{2}}}+\frac{{{x}^{2}}{{z}^{2}}}{{{y}^{4}}}+\frac{{{x}^{2}}{{y}^{2}}}{{{z}^{4}}}\]

3. সরল করি-

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

\[(ii){{a}^{2}}\left( {{b}^{2}}-{{c}^{2}} \right)+{{b}^{2}}\left( {{c}^{2}}-{{a}^{2}} \right)+{{c}^{2}}\left( {{a}^{2}}-{{b}^{2}} \right)\]

উত্তরঃ-

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

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

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

\[=\left\{ {{x}^{3}}-{{x}^{2}}y+x{{y}^{2}}+{{x}^{2}}y-x{{y}^{2}}+{{y}^{3}} \right\}+\left\{ {{x}^{3}}+{{x}^{2}}y+x{{y}^{2}}-{{x}^{2}}y-x{{y}^{2}}-{{y}^{3}} \right\}\]

\[={{x}^{3}}+{{y}^{3}}+{{x}^{3}}-{{y}^{3}}\]

\[=2{{x}^{3}}\]

\[(ii){{a}^{2}}\left( {{b}^{2}}-{{c}^{2}} \right)+{{b}^{2}}\left( {{c}^{2}}-{{a}^{2}} \right)+{{c}^{2}}\left( {{a}^{2}}-{{b}^{2}} \right)\]

\[={{a}^{2}}{{b}^{2}}-{{a}^{2}}{{c}^{2}}+{{b}^{2}}{{c}^{2}}-{{a}^{2}}{{b}^{2}}+{{a}^{2}}{{c}^{2}}-{{b}^{2}}{{c}^{2}}\]

\[=0\]

4. (i)\;a=x^2+xy+y^2,\;b=y^2+yz+z^2,\;c=z^2+xz+x^2 হলে \left(x-y\right)a\;+\;\left(y-z\right)b+\;\left(z-x\right)c –এর মান নির্ণয় করি।

(ii)\;a=lx+my+n,\;b=mx+ny+l,\;c=nx+ly+m হলে a\left(m+n\right)+b\left(n+l\right)+c(l+m) কি হয় দেখি।

উত্তরঃ-

\[(i)a={{x}^{2}}+xy+{{y}^{2}},b={{y}^{2}}+yz+{{z}^{2}},c={{z}^{2}}+xz+{{x}^{2}}\]

\[\left( x-y \right)a+\left( y-z \right)b+\left( z-x \right)c\]

\[=\left( x-y \right)\left( {{x}^{2}}+xy+{{y}^{2}} \right)+\left( y-z \right)\left( {{y}^{2}}+yz+{{z}^{2}} \right)+\left( z-x \right)\left( {{z}^{2}}+xz+{{x}^{2}} \right)\]

\[=\left\{ {{x}^{3}}+{{x}^{2}}y+x{{y}^{2}}-{{x}^{2}}y-x{{y}^{2}}-{{y}^{3}} \right\}+\left\{ {{y}^{3}}+{{y}^{2}}z+y{{z}^{2}}-{{y}^{2}}z-y{{z}^{2}}-{{z}^{3}} \right\}+\left\{ {{z}^{3}}+x{{z}^{2}}+{{x}^{2}}z-x{{z}^{2}}-{{x}^{2}}z-{{x}^{3}} \right\}\]

\[={{x}^{3}}-{{y}^{3}}+{{y}^{3}}-{{z}^{3}}+{{z}^{3}}-{{x}^{3}}\]

\[=0\]

\[(ii)a=lx+my+n,b=mx+ny+l,c=nx+ly+m\]

\[a\left( m+n \right)+b\left( n+l \right)+c\left( l+m \right)\]

\[=\left( lx+my+n \right)\left( m+n \right)+\left( mx+ny+l \right)\left( n+l \right)+\left( nx+ly+m \right)\left( l+m \right)\]

\[=mlx+{{m}^{2}}y+mn+nlx+mny+{{n}^{2}}+mnx+{{n}^{2}}y+nl+lmx+nly+{{l}^{2}}+nlx+{{l}^{2}}y+ml+mnx+mly+{{m}^{2}}\]

\[=2mlx+{{m}^{2}}y+mn+2nlx+2mny+{{n}^{2}}+{{m}^{2}}x+2ml+nly+{{l}^{2}}+{{l}^{2}}y+mnx+mly\]

কষে দেখি – 4.2

1. দুটি সংখ্যার গুণফল 3x^2+8x+4 এবং একটি সংখ্যা 3x+2 হলে, অপর সংখ্যাটি হিসাব করে লিখি।

উত্তরঃ-

দুটি সংখ্যার গুণফল =3x^2+8x+4

একটি সংখ্যা =3x+2

∴ অপর সংখ্যাটি =\left(3x^2+8x+4\right)\div\left(3x+2\right)

∴ নির্ণেয় সংখ্যাটি =(x+2)

2. একটি আয়তক্ষেত্রের ক্ষেত্রফল 24x^2-65xy+21y^2 বর্গসেমি. এবং দৈর্ঘ্য (8x -3y) সেমি. হলে প্রস্থ কত হিসাব করে লিখি।

উত্তরঃ-

আয়তক্ষেত্রটির ক্ষেত্রফল = 24x^2-65xy+21y^2  বর্গ সেমি

এবং দৈর্ঘ্য = (8x-3y) সেমি,

∴ প্রস্থ = (24x^2-65xy+21y^2)\div(8x-3y)

∴ আয়তক্ষেত্রটির প্রস্থ = (3x-7y) সেমি.

3. একটি ভাগ অঙ্কে ভাজ্য x^4+x^3y+xy^3-y^4 এবং ভাজক x^2+xy-y^2 ভাগফল ও ভাগশেষ নির্ণয় করি।

উত্তরঃ-

ভাজ্য = x^4+x^3y+xy^3-y^4=x^4+x^3y+0x^2y^2+xy^3-y^4

ভাজক = x^2+xy-y^2

∴ নির্ণেয় ভাগফল =\left(x^2+y^2\right) এবং ভাগশেষ = 0

4.  ভাগ করি-
a. \left(m^2+4m-21\right) কে \left(m-3\right) দিয়ে।
b. \left(6c^2-7c+2\right)কে \left(3c-2\right) দিয়ে।
c. \left(2a^4-a^3-2a^2+5a-1\right) কে \left(2a^2+a-3\right) দিয়ে।
d. \left(m^4-2m^3-7m^2+8m+1\right) কে \left(m^2-m-6\right) দিয়ে।

উত্তরঃ-

a)

∴ নির্ণেয় ভাগফল = \left(m+7\right) এবং ভাগশেষ = 0

b)

∴ নির্ণেয় ভাগফল = \left(2c-1\right) এবং ভাগশেষ = 0

c)

∴ নির্ণেয় ভাগফল = \left(a^2-a+1\right) এবং ভাগশেষ = \left(a+2\right)

d)

∴ নির্ণেয় ভাগফল = \left(m^2-m\right) এবং ভাগশেষ = \left(2m+12\right)

5. a) \left(6x^2a{}^3-4x^3a^2+8x^4a^2\right)\div2a^2x^2
b) \frac{2y^9x^5}{5x^2}\times\frac{125xy^5}{16x^4y^{10}}
c) \frac{7a^4y^2}{9a^2}\times\frac{729a^6}{42y^6}
d) (p^2q^2r^5-p^3q^5r^2+p^5q^3r^2)\div p^2q^2r^2

উত্তরঃ-

a)

∴ নির্ণেয় ভাগফল = \left(3a-2x+4x^2\right)

b) \[\frac{2{{y}^{9}}{{x}^{5}}}{5{{x}^{2}}}\times \frac{125x{{y}^{5}}}{16{{x}^{4}}{{y}^{10}}}\]

\[=\frac{25}{8}{{y}^{4}}\]

c) \[\frac{7{{a}^{4}}{{y}^{2}}}{9{{a}^{2}}}\times \frac{729{{a}^{6}}}{42{{y}^{6}}}\]

\[=\frac{7\times 729}{9\times 42}{{a}^{4+6-2}}{{y}^{2-6}}\]

\[=\frac{27}{2}{{a}^{8}}{{y}^{-4}}\]

\[=\frac{27}{2}\frac{{{a}^{8}}}{{{y}^{4}}}\]

d)

∴ নির্ণেয় ভাগফল = \left(r^3-pq^3+p^3q\right) এবং ভাগশেষ = 0

6. কোনো ভাগ অঙ্কে ভাজক \left(x-4\right) , ভাগফল \left(x^2+4x+4\right) ও ভাগশেষ 3 হলে ভাজ্য কত হবে হিসাব করে লিখি। [ ভাজ্য = ভাজক ×_+ ভাগশেষ ]।

উত্তরঃ-

ভাজক = \left(x-4\right), ভাগফল = \left(x^2+4x+4\right), ভাগশেষ = 3

ভাজ্য = ভাজক × ভাগফল + ভাগশেষ

\[=\left( x-4 \right)\left( {{x}^{2}}+4x+4 \right)+3\]

\[=x\left( {{x}^{2}}+4x+4 \right)-4\left( {{x}^{2}}+4x+4 \right)+3\]

\[={{x}^{3}}+4{{x}^{2}}+4x-4{{x}^{2}}-16x-16+3\]

\[={{x}^{3}}+12x-13\]

∴ নির্ণেয় ভাজ্য = x^3+12x-13

7. কোনো ভাগ অঙ্কে ভাজক \left(a^2+2a-1\right) , ভাগফল \left(5a-14\right) এবং ভাগশেষ \left(35a-17\right) হলে ভাজ্য কত হবে হিসাব করে লিখি।

উত্তরঃ-

ভাজক = \left(a^2+2a-1\right) , ভাগফল = \left(5a-14\right) এবং ভাগশেষ = \left(35a-17\right)

ভাজ্য = ভাজক × ভাগফল + ভাগশেষ

\[=\left( {{a}^{2}}+2a-1 \right)\left( 5a-14 \right)+\left( 35a-17 \right)\]

\[=\left\{ \left( {{a}^{2}}+2a-1 \right)5a-\left( {{a}^{2}}+2a-1 \right)14 \right\}+\left( 35a-17 \right)\]

\[=5{{a}^{3}}+10{{a}^{2}}-5a-14{{a}^{2}}-28a+14+35a-17\]

\[=5{{a}^{3}}-4{{a}^{2}}+2a-3\]

∴ নির্ণেয় ভাজ্য =5a^3-4a^2+2a-3

৪. ভাগ করে ভাগফল ও ভাগশেষ লিখি।
(i)\left(x^2+11x+27\right)\div\left(x+6\right)
(ii)\left(81x^4+2\right)\div\left(3x-1\right)
(iii)\left(63x^2-19x-20\right)\div\left(9x^2+5\right)
(iv)\left(x^3-x^2-8x-13\right)\div\left(x^2+3x+3\right)

উত্তরঃ-

(i)

∴ নির্ণেয় ভাগফল = \left(x+5\right) এবং ভাগশেষ = -3

(ii)

∴ নির্ণেয় ভাগফল = \left(27x^3+9x^2+3x+1\right) এবং ভাগশেষ = 3

(iii)

∴ নির্ণেয় ভাগফল = 7 এবং ভাগশেষ = -19x-35

(iv)

∴ নির্ণেয় ভাগফল = \left(x-4\right) এবং ভাগশেষ = \left(x-1\right)

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