প্রথম বীজগাণিতিক সংখ্যামালা	দ্বিতীয় বীজগাণিতিক সংখ্যামালা	\( \begin{array}{l}a^{3}+b^{3}=(a+b) \times\left(a^{2}-a b+b^{2}\right) \\ a^{3}-b^{3}=(a-b) \times\left(a^{2}+a b+b^{2}\right)\end{array} \) অভেদের সাহায্যে প্রথম ও   দ্বিতীয়ের গুণফল
(i) \(x+9\)	\(x^{2}-9 x+81\)	\((x+9)\left(x^{2}-9 x+81\right)=\left(x^{3}+9^{3}\right) =x^{3}+729\)
(ii) \(2 a-1\)	\(\frac{8 a^{3}-1}{2 a-1}=\frac{(2 a-1)\left(4 a^{2}+2 a+1\right)}{(2 a-1)} =4 a^{2}+2 a+1\)	\(8   a^{3}-1\) \(=(2   a)^{3}-(1)\) \(=(2   a-1)\left\{(2 a^{2})+2 a \times 1+(1)^{2}\right\}\) \(=(2   a-1)\left(4 a^{2}+2 a+1\right)\)
(iii) \(3-5 c\)	\(\frac{27-125 c^{3}}{3-5 c}=\frac{(3)^{3}-(5 c)^{3}}{3-5 c}\) \(=\frac{(3-5 c)\left(9+15 c+25 c^{2}\right)}{(3-5 c)}\) \(=9+15 c+25 c^{2}\)	\(27-125 c^{3}\)
(iv) \((a+b+c)\)	\((a+b)^{2}-(a+b) c+c^{2}\)	\((a+b+c)\{(a+b)^{2}-(a+b) c+c^{2} \) \(=(a+b)^{3}+(c)^{3}\)
(v) \(3 x\)	\((2 x-1)^{2}-(2 x-1)(x+1)+(x+1)^{2}\)	\( \begin{aligned} & 3 x\left\{(2 x-1)^{2}-(2 x-1)(x+1)+(x+1)^{2}\right\} \\ = & \{(2 x-1)+(x+1)\}\left\{(2 x-1)^{2}\right. \\ & \left.\quad-(2 x-1)(x+1)+(x+1)^{2}\right\} \\ = & (2 x-1)^{3}+(x+1)^{3}\end{aligned} \)
(vi) \(\frac{x}{y}+1\)	\(\frac{x^{2}}{y^{2}}-\frac{x}{y}+1\)	\(\left(\frac{x}{y}+1\right)\left(\frac{x^{2}}{y^{2}}-\frac{x}{y}+1\right)\) \( =\left(\frac{x}{y}\right)^{3}+(1)^{3}=\frac{x^{3}}{y^{3}}+1\)
(vii) \(4 a-5 b\)	\(16 a^{2}+20 a b-25 b^{2}\)	\((4 a-5 b)\left(16 a^{2}+20 a b+25 b^{2}\right)\) \(=(4 a-5 b)\left\{(4 a)^{2}+4 a \times 5 b+(5 b)^{2}\right\}\) \(=(4 a)^{3}-(5 b)^{3}\) \(=64 a^{3}-125 b^{3}\)
(viii) \( \begin{array}{l}\frac{a^{3} b^{3}-c^{3} d^{3}}{a^{2} b^{2}+a b c d+c^{2} d^{2}} \\ =\frac{(a b)^{3}-(c d)^{3}}{a^{2} b^{2}+a b c d+c^{2} d^{2}} \\ =\frac{(a b-c d)\left(a^{2} b^{2}+a b c d+c^{2} d^{2}\right)}{a^{2} b^{2}+a b c d+c^{2} d^{2}} \\ =a b-c d\end{array} \)	\(a^{2} b^{2}+a b c d+c^{2} d^{2}\)	\( a^{3} b^{3}-c^{3} d^{3} \)
(ix) \(1-4 y\)	\(\frac{1-64 y^{3}}{1-4 y}=\frac{(1)^{3}-(4 y)^{3}}{1-4 y} =\frac{(1-4 y)\left(1+4 y+16 y^{2}\right)}{(1-4 y)}\) \(=1+4 y+16 y^{2}\)	\(1-64 y^{3}\)
(x) \((2 p+1)\)	\(=\frac{8(p-3)^{3}+343}{2 p+1}\) \( =\frac{\{2(p-3)\}^{3}+(7)^{3}}{2 p+1} \) \(=\frac{(2 p-6)^{3}+(7)^{3}}{(2 p+1)}\) \(=\frac{(2 p-6+7)\left\{(2 p-6)^{2}-(2 p-6) 7+(7)^{2}\right\}}{(2 p+1)}\) \(=\frac{(2 p+1)\left\{4 p^{2}-24 p+36-14 p+42+49\right\}}{(2 p+1)}\) \(=4 p^{2}-38 p+127 \)	\(8(p-3)^{3}+343\)
(xi) \((m-p)\)	\((m+n)^{2}+(m+n)(n+p)+(n+p)^{2}\)	\( \begin{array}{l}(m-p)\left\{(m+n)^{2}+(m+n)(n+p)\right. \\ \quad \quad \quad \quad \left.+(n+p)^{2}\right\}\end{array} \) \(=\{(m+n)-(n+p)\}\left\{(m+n)^{2}\right. \left.+(m+n)(n+p)+(n+p)^{2}\right\}\) \(=(m+n)^{3}-(n+p)^{3} \)
(xii) \( \begin{array}{l}(3 a-2 b)^{2}+(3 a-2 b) \times \\ (2 a-3 b)+(2 a-3 b)^{2}\end{array} \)	\((a+b)\)	\( \begin{array}{r}(a+b)\left\{(3 a-2 b)^{2}+(3 a-2 b)(2 a-3 b)\right. \\ \left.+(2 a-3 b)^{2}\right\} \\ =\{(3 a-2 b)-(2 a-3 b)\}\{(3 a-2 b)^{2} \\ \left.\quad+(3 a-2 b)(2 a-3 b)+(2 a-3 b)^{2}\right\} \\ =(3 a-2 b)^{3}-(2 a-3 b)^{3}\end{array} \)

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

\( (a-2 b)\left(a^{2}+2 a b+4 b^{2}\right)\left(a^{3}+8 b^{3}\right)\)
\(=\left\{(a)^{3}-(2 b)^{3}\right\}\left(a^{3}+8 b^{3}\right) \)
\( \left[\because a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\right] \)
\( =\left(a^{3}-8 b^{3}\right)\left(a^{3}+8 b^{3}\right)\)
\(=\left(a^{3}\right)^{2}-\left(8 b^{3}\right)^{2}\) \( \left[\because a^{2}-b^{2}=(a-b)(a+b)\right] \)
\(=a^{6}-64 b^{6} \)

\( \left(4 a^{2}-9\right)\left(4 a^{2}-6 a+9\right)\left(4 a^{2}+6 a+9\right)\)
\(=\left\{(2 a)^{2}-(3)^{2}\right\}\left(4 a^{2}-6 a+9\right)\left(4 a^{2}+6 a+9\right)\)
\(=(2 a-3)(2 a+3)\left(4 a^{2}-6 a+9\right)\left(4 a^{2}+6 a+9\right) \)
\( \left[\because a^{2}-b^{2}=(a+b)(a-b)\right] \)
\( =(2 a+3)\left(4 a^{2}-6 a+9\right)(2 a-3)\left(4 a^{2}+6 a+9\right) \)
\( \left.=\left\{(2 a)^{3}+(3)^{3}\right\}(2 a)^{3}-(3)^{3}\right\} \)
[\( \because(a+b)\left(a^{2}-a b+b^{2}\right)=a^{3}+b^{3} \)
\( (a-b)\left(a^{2}+a b+b^{2}\right)=a^{3}-b^{3} \)]
\( =\left(8 a^{3}+27\right)\left(8 a^{3}-27\right)\)
\(=\left(8 a^{3}\right)^{2}-(27)^{2} \quad\left[\because(a+b)(a-b)=a^{2}-b^{2}\right]\)
\(=64 a^{6}-729 \)

\( (x-y)\left(x^{2}+x y+y^{2}\right)+(y-z)\left(y^{2}+y z+z^{2}\right)\)
\(\quad \quad \quad \quad \quad +(z-x)\left(z^{2}+z x+x^{2}\right) \)
\( =x^{3}-y^{3}+y^{3}-z^{3}+z^{3}-x^{3} \)\( \quad \quad \quad \quad \quad \quad \quad \left[\because(a-b)\left(a^{2}+a b+b^{2}\right)=a^{3}-b^{3}\right] \)
\(=0\)

\( (x+1)\left(x^{2}-x+1\right)+(2 x-1)\left(4 x^{2}+2 x+1\right) \)
\( \quad \quad \quad \quad \quad \quad \quad -(x-1)\left(x^{2}+x+1\right) \)
\( =x^{3}+1^{3}+(2 x)^{3}-(1)^{3}-(x)^{3}+(1)^{3} \)\( \quad \quad \quad \quad \quad \quad \quad \quad \quad \left[\because(a+b)\left(a^{2}-a b+b^{2}\right)=a^{3}+b^{3}\right] \)
\( =x^{3}+1+8 x^{3}-1-x^{3}+1\)
\(=8 x^{3}+1 \)

\(x+\frac{1}{x}=-1\)
বা, \(\left(x+\frac{1}{x}\right) x=-1(x)\)
বা, \(x^{2}+1=-x\)
বা, \(x^{2}+x+1=0\) [পক্ষান্তরে পাই]
\(x^{3}-1\)
= \((x-1) \times\left(x^{2}+x+1\right)\) [ সূত্রের সাহায্যে ]
= \((x-1) \times 0=0\)

প্রদত্ত, \( a+\frac{9}{a}=3 \)
বা, \( \frac{a^{2}+9}{a}=3 \)
বা, \( a^{2}+9=3 a \)
\( \therefore a^{2}-3 a+9=0 \)
প্রদত্ত রাশিমালা, \( a^{3}+27 \)
\( =(a)^{3}+(3)^{3}=(a+3)\left\{a^{2}-3 \cdot a+(3)^{2}\right\} \)\( \left[\because a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\right] \)
\( =(a+3)\left(a^{2}-3 a+9\right)\)
\(=(a+3) \times 0\) \( \quad \left[\because a^{2}-3 a+9=0\right] \)
\(=0 \)
\( \therefore a^{3}+27=0 \)

\( \frac{a}{b}+\frac{b}{a}=1 \quad \)
বা, \( \frac{a^{2}+b^{2}}{a b}=1 \)
বা, \( a^{2}+b^{2}=a b \)
বা, \( a^{2}-a b+b^{2}=0 \)
\(\therefore\) প্রদত্ত রাশিমালা :
\( a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right) \)
\( =(a+b) \times 0 \) \( \quad \left[\because a^{2}-a b+b^{2}=0\right] \)
\(= 0\)
\( \therefore a^{3}+b^{3}=0 \)

\(1000 a^{3}+27 b^{6}\)
\( =(10 a)^{3}+\left(3 b^{2}\right)^{3}\)
\(=\left(10 a+3 b^{2}\right)\left\{(10 a)^{2}-10 a \cdot 3 b^{2}+\left(3 b^{2}\right)^{2}\right\}\)
\(=\left(10 a+3 b^{2}\right)\left(100 a^{2}-30 a b^{2}+9 b^{4}\right) \)

\(1-216 z^{3}\)
\(=(1)^{3}-(6 z)^{3}\)
\( =(1-6 z)\left\{1^{2}+1.6 z+(6 z)^{2}\right\}\)
\(=(1-6 z)\left(1+6 z+36 z^{2}\right) \)

\(m^{4}-m\)
\( =m\left(m^{3}-1\right) \)
\( =m\left\{(m)^{3}-(1)^{3}\right\}\)
\(=m(m-1)\left(m^{2}+m .1+1^{2}\right)\)
\(=m(m-1)\left(m^{2}+m+1\right) \)

\(192 a^{3}+3\)
\( =3\left(64 a^{3}+1\right) \)
\(=3\left\{(4 a)^{3}+(1)^{3}\right\}\)
\(=3(4 a+1)\left\{(4 a)^{2}-4 a .1+(1)^{2}\right\}\)
\(=3(4 a+1)\left(16 a^{2}-4 a+1\right) \)

\(16 a^{4} x^{3}+54 a y^{3}\)
\( =2 a\left(8 a^{3} x^{3}+27 y^{3}\right)\)
\(=2 a\left\{(2 a x)^{3}+(3 y)^{3}\right\}\)
\(=2 a(2 a x+3 y)\left\{(2 a x)^{2}-2 a x \cdot 3 y+(3 y)^{2}\right\}\)
\(=2 a(2 a x+3 y)\left(4 a^{2} x^{2}-6 a x y+9 y^{2}\right) \)

\(729 a^{3} b^{3} c^{3}-125\)
\( =(9 a b c)^{3}-(5)^{3}\)
\(=(9 a b c-5)\left\{(9 a b c)^{2}+9 a b c .5+5^{2}\right\}\)
\(=(9 a b c-5)\left(81 a^{2} b^{2} c^{2}+45 a b c+25\right) \)

\( \frac{27}{a^{3}}-\frac{1}{27 b^{3}}\)
\(=\left(\frac{3}{a}\right)^{3}-\left(\frac{1}{3 b}\right)^{3} \)
\(=\left(\frac{3}{a}-\frac{1}{3 b}\right)\left\{\left(\frac{3}{a}\right)^{2}+\frac{3}{a} \times \frac{1}{3 b}+\left(\frac{1}{3 b}\right)^{2}\right\}\)
\(=\left(\frac{3}{a}-\frac{1}{3 b}\right)\left\{\frac{9}{a^{2}}+\frac{1}{a b}+\frac{1}{9 b^{2}}\right\} \)

\(\frac{x^{3}}{64}-\frac{64}{x^{3}}\)
\( =\left(\frac{x}{4}\right)^{3}-\left(\frac{4}{x}\right)^{3}\)
\(=\left(\frac{x}{4}-\frac{4}{x}\right)\left\{\left(\frac{x}{4}\right)^{2}+\frac{x}{4} \cdot \frac{4}{x}+\left(\frac{4}{x}\right)^{2}\right\}\)
\(=\left(\frac{x}{4}-\frac{4}{x}\right)\left\{\left(\frac{x}{4}\right)^{2}+\left(\frac{4}{x}\right)^{2}+1\right\}\)
\(=\left(\frac{x}{4}-\frac{4}{x}\right)\left\{\left(\frac{x}{4}+\frac{4}{x}\right)^{2}-2 \cdot \frac{x}{4} \cdot \frac{4}{x}+1\right\}\)
\(=\left(\frac{x}{4}-\frac{4}{x}\right)\left\{\left(\frac{x}{4}+\frac{4}{x}\right)^{2}-1\right\}\)
\(=\left(\frac{x}{4}-\frac{4}{x}\right)\left\{\left(\frac{x}{4}+\frac{4}{x}\right)^{2}-(1)^{2}\right\}\)
\(=\left(\frac{x}{4}-\frac{4}{x}\right)\left(\frac{x}{4}+\frac{4}{x}+1\right)\left(\frac{x}{4}+\frac{4}{x}-1\right) \)

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

\( 1+9 x+27 x^{2}+28 x^{3}\)
\(=1+9 x+27 x^{2}+27 x^{3}+x^{3}\)
\(=1+3 \cdot 1^{2} \cdot 3 x+3 \cdot 1(3 x)^{2}+(3 x)^{3}+x^{3}\)
\(=(1+3 x)^{3}+(x)^{3}\)
\(=\{(1+3 x)+x\}\left\{(1+3 x)^{2}-(1+3 x) \cdot x+(x)^{2}\right\}\)
\(=(1+4 x)\left(1+6 x+9 x^{2}-x-3 x^{2}+x^{2}\right)\)
\(=(1+4 x)\left(1+5 x+7 x^{2}\right) \)

\( x^{3}-9 y^{3}-3 x y(x-y)\)
\(=x^{3}-9 y^{3}-3 x^{2} y+3 x y^{2}\)
\(=x^{3}-y^{3}-8 y^{3}-3 x^{2} y+3 x y^{2}\)
\(=\left(x^{3}-3 x^{2} y+3 x y^{2}-y^{3}\right)-8 y^{3}\)
\(=(x-y)^{3}-(2 y)^{3}\)
\( =\{(x-y)-2 y\}\left\{(x-y)^{2}+(x-y) \cdot 2 y+(2 y)^{2}\right\} \)
\(=(x-3 y)\left(x^{2}-2 x y+y^{2}+2 x y-2 y^{2}+4 y^{2}\right)\)
\(=(x-3 y)\left(x^{2}+3 y^{2}\right) \)

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

\( x^{6}+3 x^{4} b^{2}+3 x^{2} b^{4}+b^{6}+a^{3} b^{3}\)
\(=\left(x^{2}\right)^{3}+3 \cdot\left(x^{2}\right)^{2} \cdot b^{2}+3 x^{2} \cdot\left(b^{2}\right)^{2}+\left(b^{2}\right)^{3}+(a b)^{3}\)
\(=\left(x^{2}+b^{2}\right)^{3}+(a b)^{3}\)
\(=\left(x^{2}+b^{2}+a b\right)\left\{\left(x^{2}+b^{2}\right)^{2}-\left(x^{2}+b^{2}\right) a b+(a b)^{2}\right\}\)
\(=\left(x^{2}+b^{2}+a b\right) \)
\( \quad \quad \quad \quad \left\{x^{4}+2 x^{2} b^{2}+b^{4}-a b\left(x^{2}+b^{2}\right)+a^{2} b^{2}\right\} \)

\( x^{6}+27\)
\(=\left(x^{2}\right)^{3}+(3)^{3}\)
\(=\left(x^{2}+3\right)\left\{\left(x^{2}\right)^{2}-x^{2} \cdot 3+(3)^{2}\right\}\)
\(=\left(x^{2}+3\right)\left(x^{4}-3 x^{2}+9\right)\)
\(=\left(x^{2}+3\right)\left\{\left(x^{2}\right)^{2}+(3)^{2}-3 x^{2}\right\}\)
\(=\left(x^{2}+3\right)\left\{\left(x^{2}+3\right)^{2}-2 \cdot x^{2} \cdot 3-3 x^{2}\right\}\)
\(=\left(x^{2}+3\right)\left\{\left(x^{2}+3\right)^{2}-9 x^{2}\right\}\)
\(=\left(x^{2}+3\right)\left\{\left(x^{2}+3\right)^{2}-(3 x)^{2}\right\}\)
\(=\left(x^{2}+3\right)\left(x^{2}+3+3 x\right)\left(x^{2}+3-3 x\right) \)

\( x^{6}-y^{6}=\left(x^{3}\right)^{2}-\left(y^{3}\right)^{2} \)
\( =\left(x^{3}+y^{3}\right)\left(x^{3}-y^{3}\right)\)
\(=(x+y)\left(x^{2}-x y+y^{2}\right)(x-y)\left(x^{2}+x y+y^{2}\right) \)

\( x^{12}-y^{12}\)
\(=\left(x^{6}\right)^{2}-\left(y^{6}\right)^{2} \)
\( =\left(x^{6}+y^{6}\right)\left(x^{6}-y^{6}\right)\)
\(=\left\{\left(x^{3}\right)^{2}-\left(y^{3}\right)^{2}\right\}\left\{\left(x^{2}\right)^{3}+\left(y^{2}\right)^{3}\right\}\)
\(=\left(x^{3}-y^{3}\right)\left(x^{3}+y^{3}\right)\left\{\left(x^{2}\right)^{3}+\left(y^{2}\right)^{3}\right\}\)
\(=(x-y)\left(x^{2}+x y+y^{2}\right)(x+y)\left(x^{2}-x y+y^{2}\right)\)
\(\quad \quad \left(x^{2}+y^{2}\right)\left\{\left(x^{2}\right)^{2}-x^{2} y^{2}+\left(y^{2}\right)^{2}\right\} \)
\( =(x-y)\left(x^{2}+x y+y^{2}\right)(x+y)\left(x^{2}-x y+y^{2}\right) \)
\(\quad \quad \quad \left(x^{2}+y^{2}\right)\left(x^{4}-x^{2} y^{2}+y^{4}\right) \)

\( m^{3}-n^{3}-m\left(m^{2}-n^{2}\right)+n(m-n)^{2}\)
\(=(m-n)\left(m^{2}+m n+n^{2}\right)-m(m+n)(m-n)+n(m-n)(m-n) \)
\( =(m-n)\left\{m^{2}+m n+n^{2}-m(m+n)+n(m-n)\right\}\)
\(=(m-n)\left\{m^{2}+m n+n^{2}-m^{2}-m n+m n-n^{2}\right\}\)
\(=(m-n) m n\)
\(=m n(m-n) \)