প্রথম বীজগাণিতিক সংখ্যামালা	দ্বিতীয় বীজগাণিতিক সংখ্যামালা	গুণফল	গুণফলের মান
(a) \(x^{2}-3 x+5\)	\(5 x+9\)	\(5 x^{3}-6 x^{2}-2 x+45\)z	\(x=1\) বসিয়ে পেলাম 42
(b) \(x^{2}+12-7 y\)	\(2 z-y\)	\(2 x^{3}-x^{2} y-14 x y+24 x-12 y-7 y^{2}\)	\(x=-2\) ও y = 2 বসিয়ে পেলাম –12
(c) \(8 p^{3}-3 p-2 p^{2}\)	\(4 p^{2}-5\)	\(32 p^{5}-8 p^{4}-52 p^{3}+10 p^{2}+15 p\)	\(p=-2\) বসিয়ে পেলাম 198
(d) \(6 a+5 b+2\)	\(a-b+6\)	\(6 a^{2}+38 a-a b+28 b-5 b^{2}+12\)	a = 0 ও b = –1 বসিয়ে পেলাম –21
(e) \(p^{3}-p^{2} q^{2}+q^{3}\)	\(p^{2}+p q+q^{2}\)	\(p^{5}+p^{4} q-p^{4} q^{2}-p^{3} q^{3}+p^{3} q^{2}-p^{2} q^{4}+p^{2} q^{3}+p q^{4}+q^{5}\)	p = 2 ও q = –2 বসিয়ে পেলাম –64
(f) \(x^{2}+y^{2}+z^{2}-x y-y z-z x\)	\(x+y+z\)	\(x^{2}-3 x y z+y^{3}+y^{3}\)	\(x=1, y=0, z=-1\) বসিয়ে পেলাম 0
(g) \( a^{2}+b^{2} \)	\( (a+b)(a-b)\left(a^{4}+b^{4}\right) \)	\( a^{8}-b^{8} \)	\( a=2, b=2 \) বসিয়ে পেলাম \(0\)

(b) \( 2 x^{3}-x^{2} y+24 x-12 y-14 x y+7 y^{2}\)
\(=2 \times(-2)^{3}-(-2)^{2} \times 2+24 \times(-2)-12 \times 2 -14 \times(-2) \times 2+7(2)^{2} \)[\( x=-2, y=2 \) বসিয়ে পাই]
\( =2 \times(-8)-4 \times 2+(-48)-24+56+28\)
\(=-16-8-48-24+56+28\)
\(=-96+84=-12 \)
(c) \( 32 p^{5}-8 p^{4}-52 p^{3}+10 p^{2}+15 p\)
\(=32 \times(-2)^{5}-8 \times(-2)^{4}-52 \times(-2)^{3}+10 \times(-2)^{2}\)
\(+15 \times(-2)\) [\(p=-2\) বসিয়ে পাই]
\(=32 \times(-32)-8 \times 16-52 \times(-8)+10 \times 4-30\)
\(=-1024-128+416+40-30\)
\(=-1182+456=-726 \)
(d) \( 6 a^{2}-a b+38 a+28 b-5 b^{2}+12\)
\(=6 \times(0)^{2}-0 \times(-1)+38 \times 0+28 \times(-1)-5(-1)^{2}+12 \)[ \(a=0\) ও \(b=-1\) বসিয়ে পাই]
\( =0-0+0-28-5+12=-33+12=-21 \)
(e) \( p^{5}-p^{4} q^{2}+p^{4} q-p^{3} q^{3}+p q^{4}+p^{3} q^{2}+p^{2} q^{3}-p^{2} q^{4}+q^{5}\)
\(=(2)^{5}-(2)^{4}(-2)^{2}+(2)^{4}(-2)-(2)^{3}(-2)^{3}+2 \times(-2)^{4} \) \(\quad \quad +(2)^{3}(-2)^{2}+(2)^{2}(-2)^{3}-(2)^{2}(-2)^{4}+(-2)^{5} \)
[ \( p=2, q=-2 \) বসিয়ে পাই]
\( =32-16 \times 4+16 \times(-2)-8 \times(-8)+2 \times 16\) \(\quad \quad \quad +8 \times 4+4 \times(-8)-4 \times(+16)+(-32) \)
\(=32-64-32+64+32+32-32-64-32 \)
\(=-64\)
(f) \( x^{3}+y^{3}+z^{3}-3 x y z\)
\(=(1)^{3}+(0)^{3}+(-1)^{3}-3 \times 1 \times 0 \times(-1) \)
[ \( x=1, y=0, z=-1 \) বসিয়ে পাই]
\( =1+0-1-0=0 \)
(g) \( a^{8}-b^{8}=(2)^{8}-(2)^{8} \) [\(a=2\) এবং \(b=2\) বসিয়ে পাই]
\( =256-256=0 \)

\( \left(x^{5}+1\right) \times\left(3-x^{4}\right) \times\left(4+x^{3}+x^{6}\right)\)
\(=\left(3 x^{5}+3-x^{9}-x^{4}\right) \times\left(4+x^{3}+x^{6}\right)\)
\(=\left(-x^{9}+3 x^{5}-x^{4}+3\right) \times\left(x^{6}+x^{3}+4\right) \)
[ ঘাতের নিম্নক্রমে সাজিয়ে পাই ]
\( =-x^{9}\left(x^{6}+x^{3}+4\right)+3 x^{5}\left(x^{6}+x^{3}+4\right)-x^{4}\left(x^{6}+x^{3}+4\right)+3\left(x^{6}+x^{3}+4\right) \)
\( =-x^{15}-x^{12}-4 x^{9}+3 x^{11}+3 x^{8}+12 x^{5}-x^{10}-x^{7}-4 x^{4}+3 x^{6}+3 x^{3}+12 \)
\( =-x^{15}-x^{12}+3 x^{11}-x^{10}-4 x^{9}+3 x^{8}-x^{7}+3 x^{6}+12 x^{5}-4 x^{4}+3 x^{3}+12 \)

\( \left(2 a^{3}-3 b^{5}\right) \times\left(2 a^{3}+3 b^{5}\right) \times\left(2 a^{4}-3 a^{2} b^{2}+b^{4}\right)\)
\(=\left\{\left(2 a^{3}-3 b^{5}\right) 2 a^{3}+\left(2 a^{3}-3 b^{5}\right) 3 b^{5}\right\}\left(2 a^{4}-3 a^{2} b^{2}+b^{4}\right)\)
\(=\left\{4 a^{6}-6 a^{3} b^{5}+6 a^{3} b^{5}-9 b^{10}\right\}\left(2 a^{4}-3 a^{2} b^{2}+b^{4}\right) \)
\( =\left(4 a^{6}-9 b^{10}\right)\left(2 a^{4}-3 a^{2} b^{2}+b^{4}\right)\)
\(=4 a^{6}\left(2 a^{4}-3 a^{2} b^{2}+b^{4}\right)-9 b^{10}\left(2 a^{4}-3 a^{2} b^{2}+b^{4}\right)\)
\(=8 a^{10}-12 a^{8} b^{2}+4 a^{6} b^{4}-18 a^{4} b^{10}+27 a^{2} b^{12}-9 b^{14} \)

\( (a x+b y) \times(a x-b y) \times\left(a^{4} x^{4}+a^{2} b^{2} x^{2} y^{2}+b^{4} y^{4}\right) \)
\( =\left(a^{2} x^{2}-a x b y+a x b y-b^{2} y^{2}\right)\left(a^{4} x^{4}+a^{2} b^{2} x^{2} y^{2}+b^{4} y^{4}\right) \)
\( =\left(a^{2} x^{2}-b^{2} y^{2}\right)\left(a^{4} x^{4}+a^{2} b^{2} x^{2} y^{2}+b^{4} y^{4}\right)\)
\(=a^{2} x^{2}\left(a^{4} x^{4}+a^{2} b^{2} x^{2} y^{2}+b^{4} y^{4}\right)-b^{2} y^{2}\left(a^{4} x^{4}+a^{2} b^{2} x^{2} y^{2}+b^{4} y^{4}\right) \)
\( =a^{6} x^{6}+a^{4} b^{2} x^{4} y^{2}+a^{2} b^{4} x^{2} y^{4}-a^{4} b^{2} x^{4} y^{2}-a^{2} b^{4} x^{2} y^{4}-b^{6} y^{6}=a^{6} x^{6}-b^{6} y^{6} \)

\( (a+b+c) \times(a-b+c) \times(a+b-c) \)
\( =\{a(a-b+c)+b(a-b+c)+c(a-b+c)\} \times (a+b-c) \)
\( =\left(a^{2}-a b+a c+a b-b^{2}+b c+a c-b c+c^{2}\right) (a+b-c) \)
\( =\left(a^{2}+2 a c-b^{2}+c^{2}\right)(a+b-c)\)
\(=a^{2}(a+b-c)+2 a c(a+b-c)-b^{2}(a+b-c)+c^{2}(a+b-c) \)
\( =a^{3}+a^{2} b-a^{2} c+2 a^{2} c+2 a b c-2 a c^{2}-a b^{2}-b^{3}+b^{2} c+a c^{2}+b c^{2}-c^{3} \)
\( =a^{3}-b^{3}-c^{3}+a^{2} b-a b^{2}+a^{2} c-a c^{2}+b^{2} c+b c^{2}+2abc\)

\( \left(\frac{2 p^{2}}{q^{2}}+\frac{5 q^{2}}{p^{2}}\right) \times\left(\frac{2 p^{2}}{q^{2}}-\frac{5 q^{2}}{p^{2}}\right)\)
\(=\frac{2 p^{2}}{q^{2}}\left(\frac{2 p^{2}}{q^{2}}-\frac{5 q^{2}}{p^{2}}\right)+\frac{5 q^{2}}{p^{2}}\left(\frac{2 p^{2}}{q^{2}}-\frac{5 q^{2}}{p^{2}}\right)\)
\(=\frac{4 p^{4}}{q^{4}}-\frac{10 p^{2} q^{2}}{p^{2} q^{2}}+\frac{10 p^{2} q^{2}}{p^{2} q^{2}}-\frac{25 q^{4}}{p^{4}}=\frac{4 p^{4}}{q^{4}}-\frac{25 q^{4}}{p^{4}} \)

\( \left(\frac{x^{2}}{y^{2}}+\frac{y^{2}}{z^{2}}\right) \times\left(\frac{y^{2}}{z^{2}}+\frac{z^{2}}{x^{2}}\right) \times\left(\frac{z^{2}}{x^{2}}+\frac{x^{2}}{y^{2}}\right)\)
\(=\left\{\left(\frac{x^{2}}{y^{2}}+\frac{y^{2}}{z^{2}}\right) \frac{y^{2}}{z^{2}}\right\}+\left\{\left(\frac{x^{2}}{y^{2}}+\frac{y^{2}}{z^{2}}\right) \frac{z^{2}}{x^{2}}\right\} \times\left(\frac{z^{2}}{x^{2}}+\frac{x^{2}}{y^{2}}\right)\)
\(=\left\{\frac{x^{2}}{y^{2}} \times \frac{y^{2}}{z^{2}}+\frac{y^{2}}{z^{2}} \times \frac{y^{2}}{z^{2}}+\frac{x^{2}}{y^{2}} \times \frac{z^{2}}{x^{2}}+\frac{y^{2}}{z^{2}} \times \frac{z^{2}}{x^{2}}\right\} \times \left(\frac{z^{2}}{x^{2}}+\frac{x^{2}}{y^{2}}\right) \)
\( =\left(\frac{x^{2}}{z^{2}}+\frac{y^{4}}{z^{4}}+\frac{z^{2}}{y^{2}}+\frac{y^{2}}{x^{2}}\right)\left(\frac{z^{2}}{x^{2}}+\frac{x^{2}}{y^{2}}\right)\)
\(=\left(\frac{x^{2}}{z^{2}}+\frac{z^{2}}{y^{2}}+\frac{y^{4}}{z^{4}}+\frac{y^{2}}{x^{2}}\right) \times \frac{z^{2}}{x^{2}}+ \left(\frac{x^{2}}{z^{2}}+\frac{z^{2}}{y^{2}}+\frac{y^{4}}{z^{4}}+\frac{y^{2}}{x^{2}}\right) \times \frac{x^{2}}{y^{2}} \)
\( =\frac{x^{2}}{z^{2}} \times \frac{z^{2}}{x^{2}}+\frac{z^{2}}{y^{2}} \times \frac{z^{2}}{x^{2}}+\frac{y^{4}}{z^{4}} \times \frac{z^{2}}{x^{2}}+\frac{y^{2}}{x^{2}} \times \frac{z^{2}}{x^{2}}+\frac{x^{2}}{z^{2}} \times \frac{x^{2}}{y^{2}}+\frac{z^{2}}{y^{2}} \times \frac{x^{2}}{y^{2}}+\frac{y^{4}}{z^{4}} \times \frac{x^{2}}{y^{2}}+\frac{y^{2}}{x^{2}} \times \frac{x^{2}}{y^{2}} \)
\( =1+\frac{z^{4}}{x^{2} y^{2}}+\frac{y^{4}}{x^{2} z^{2}}+\frac{y^{2} z^{2}}{x^{4}}+\frac{x^{4}}{y^{2} z^{2}}+\frac{x^{2} z^{2}}{y^{4}}+\frac{x^{2} y^{2}}{z^{4}}+1\)
\(=2+\frac{x^{4}}{y^{2} z^{2}}+\frac{y^{4}}{x^{2} z^{2}}+\frac{z^{4}}{x^{2} y^{2}}+\frac{y^{2} z^{2}}{x^{4}}+\frac{z^{2} x^{2}}{y^{4}}+\frac{x^{2} y^{2}}{z^{4}} \)

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

\( a^{2}\left(b^{2}-c^{2}\right)+b^{2}\left(c^{2}-a^{2}\right)+c^{2}\left(a^{2}-b^{2}\right)\)
\(=a^{2} b^{2}-a^{2} c^{2}+b^{2} c^{2}-b^{2} a^{2}+c^{2} a^{2}-c^{2} b^{2}=0 \)

প্রদত্ত সংখ্যামালা,
\( (x-y) a+(y-z) b+(z-x) c \)
\( =(x-y)\left(x^{2}+x y+y^{2}\right)+(y-z)\left(y^{2}+y z+z^{2}\right)+(z-x)\left(z^{2}+x z+x^{2}\right) \)[a, b ও c এর মান বসিয়ে পাই]
\( =x\left(x^{2}+x y+y^{2}\right)-y\left(x^{2}+x y+y^{2}\right)+y\left(y^{2}+y z+z^{2}\right)\)
\(\quad \quad \quad \quad -z\left(y^{2}+y z+z^{2}\right)+z\left(z^{2}+x z+x^{2}\right)-x\left(z^{2}+x z+x^{2}\right)\)
\( \begin{array}{c}=x^{3}+x^{2} y+x y^{2}-x^{2} y-x y^{2}-y^{3}+y^{3}+y^{2} z+y z^{2} \\ -y^{2} z-y z^{2}-z^{2}+z^{3}+x z^{2}+x^{2} z \\ -x z^{2}-x^{2} z-x^{3}\end{array} \) \(=0\)

\( a(m+n)+b(n+l)+c(l+m)\)
\(=(l x+m y+n)(m+n)+(m x+n y+l)(\mathrm{n}+l) \) \(\quad \quad +(n x+l y+m)(l+m)\quad \) [\(a, b\) ও \(c\)-এর মান বসিয়ে পাই]
\( =m l x+m^{2} y+m n+n l x+m n y+n^{2}+m n x+n^{2} y+n l \) \(\quad \quad +\mathrm{m} l x+\mathrm{n} l y+l^{2}+\mathrm{n} l x+l^{2} \mathrm{y}+\mathrm{m} l+\mathrm{mn} x+\mathrm{m} l \mathrm{y}+\mathrm{m}^{2} \)
\( =\mathrm{m}^{2}+\mathrm{n}^{2}+l^{2}+l \mathrm{~m}+l \mathrm{n}+\mathrm{mn}+\mathrm{y}\left(l^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}\right. \) \(\quad \quad +l m+m n+l n)+2 m l x+2 \mathrm{n} l x+2 \mathrm{mn} x \)
\( =\mathrm{m}^{2}+\mathrm{n}^{2}+l^{2}+l \mathrm{~m}+l \mathrm{n}+\mathrm{mn}\) \(\quad \quad +y\left(l^{2}+m^{2}+n^{2}+l m+l n+m n\right)+2 x(m l+l n+m n) \)