উৎপাদকে বিশ্লেষণ কর :

$$\begin{gathered} a^3+8 \\ ={\left(a\right)}^3+{\left(2\right)}^3=(a+2)\left\{{\left(a\right)}^2-a.2+{\left(2\right)}^2\right\} \\ =(a+2)(a^2-2a+4) \end{gathered}$$

$$\begin{gathered} 8x^3+343 \\ ={\left(2x\right)}^3+{\left(7\right)}^3=(2x+7)\left\{{\left(2x\right)}^2-2x.7+{\left(7\right)}^2\right\} \\ =(2x+7)(4x^2-14x+49) \end{gathered}$$

$$\begin{gathered} 8a^4+27a{b}^3 \\ =a(8a^3+27b^3)=a\left\{\right(2a{)}^3+\left(3b{)}^3\right\} \\ =a(2a+3b)\left\{\right(2a{)}^2-2a.3b+\left(3b{)}^2\right\} \\ =a(2a+3b)(4a^2-6ab+9b^2) \end{gathered}$$

$$\begin{gathered} 8x^3+1 \\ =(2x{)}^3+(1{)}^3 \end{gathered}$$

$$\begin{gathered} =(2x+1)\left\{\right(2x{)}^2-2x.1+\left(1{)}^2\right\} \\ =(2x+1)(4x^2-2x+1) \end{gathered}$$

$64a^3-1525b^3$

$={\left(4a\right)}^3-{\left(5b\right)}^3$

$=(4a-5b)\left\{\right(4a{)}^2+4a.5b+\left(5b{)}^2\right\}$

$=(4a-5b)(16a^2+20ab+25b^2)$

$$\begin{gathered} 729a^3-64b^3{c}^6 \\ =(9a{)}^3-(4b{c}^2{)}^3 \end{gathered}$$

$$\begin{gathered} =(9a-4b{c}^2)\left\{\right(9a{)}^2+9a.4b{c}^2+\left(4b{c}^2{)}^2\right\} \\ =(9a-4b{c}^2)(81a^2+36ab{c}^2+16b^2{c}^4) \end{gathered}$$

$27a^3{b}^3+64b^3{c}^3$

$=(9a{)}^3-(4b{c}^2{)}^3$

$$\begin{gathered} =b^3(27a^3+64c^3) \\ =b^3\left\{\right(3a{)}^3+\left(4c{)}^3\right\} \end{gathered}$$

$$\begin{gathered} =b^3(3a+4c)\left\{\right(3a{)}^2-3a.4c+\left(4c{)}^2\right\} \\ =b^3(3a+4c)(9a^2-12ac+16c^2) \end{gathered}$$

$$\begin{gathered} 56x^3-189y^3 \\ =7 (8x^3-27y^3) \\ =7\left\{{\left(2x\right)}^3+2x.3y+{\left(3y\right)}^3\right\} \end{gathered}$$

$$\begin{gathered} 3x-75x^3 \\ =3x(1-25x^2) \end{gathered}$$

$$\begin{gathered} =3x\left\{\right(1{)}^2-\left(5x{)}^2\right\} \\ = 3x(1 + 5x)(1 - 5x) \end{gathered}$$

$$\begin{gathered} 4x^2-y^2 \\ =(2x{)}^2-(y{)}^2 \end{gathered}$$

$=(2x+y)(2x-y)$

$$\begin{gathered} 3a{y}^2-48a \\ =3a(y^2-16) \end{gathered}$$

$$\begin{gathered} =3a\left\{\right(y{)}^2-\left(4{)}^2\right\} \\ = 3a(y + 4)(y - 4) \end{gathered}$$

$$\begin{gathered} a^2-2ab+b^2-p^2 \\ =(a^2-2ab+b^2)-p^2 \\ =(a-b{)}^2-(p{)}^2 \end{gathered}$$

$=(a-b+p)(a-b-p)$

$$\begin{gathered} 16y^2-a^2-6a-9 \\ =16y^2-(a^2+6a+9) \end{gathered}$$

$$\begin{gathered} =16y^2- \left\{(a{)}^2+2.a.3+(3{)}^2\right\} \\ =(4y{)}^2-(a+3{)}^2 \end{gathered}$$

$= (4y+a+3)(4y-a-3)$

$$\begin{gathered} 8a+a{p}^3 \\ =a(8+p^3) \\ =a\left\{\right(2{)}^3+\left(p{)}^3\right\} \end{gathered}$$

$$\begin{gathered} =a(2+p) \left\{(2{)}^2-2.p+(p{)}^2\right\} \\ =a(2+p)(4-2p+p^2) \end{gathered}$$

$$\begin{gathered} 2a^3+16b^3 \\ =2(a^3+8b^3) \\ =2\left\{\right(a{)}^3+\left(2b{)}^3\right\} \end{gathered}$$

$$\begin{gathered} =2(a+2b)\left\{\right(a{)}^2-a.2b+\left(2b{)}^2\right\} \\ =2(a+2b)(a^2-2ab+4b^3) \end{gathered}$$

$$\begin{gathered} x^2+y^2-2xy-1 \\ =(x^2-2xy+y^2)-1 \end{gathered}$$

$$\begin{gathered} =(x-y{)}^2-(1{)}^2 \\ =(x-y+1)(x-y-1) \end{gathered}$$

$$\begin{gathered} a^2-2ab+2b-1 \\ =a^2-1-2ab+2b \end{gathered}$$

$$\begin{gathered} =(a+1)(a-1)-2b(a-1) \\ =(a-1)(a-2b+1) \end{gathered}$$

$$\begin{gathered} x^4-2x^2+1 \\ =(x^2{)}^2-2.x^2.1+{\left(1\right)}^2 \end{gathered}$$

$=(x^2-1{)}^2$

$=\left\{\right(x+1\left)\right(x-1){\}}^2$

$=(x+1{)}^2(x-1{)}^2$

$$\begin{gathered} 36-12x+x^2 \\ =(6{)}^2-2.6.x+(x{)}^2 \end{gathered}$$

$$\begin{gathered} =(6-x{)}^2 \\ =\{-(x-6){\}}^2 \end{gathered}$$

$=(x-6{)}^2$

$$\begin{gathered} x^6-y^6 \\ =(x^3{)}^2-(y^3{)}^2 \end{gathered}$$

$$\begin{gathered} =(x^3+y^3)(x^3-y^3) \\ =(x+y)(x^2-xy+y^2)(x-y)(x^2+xy+y^2) \end{gathered}$$

$=(x+y)(x-y)(x^2-xy+y^2)(x^2+xy+y^2)$

$$\begin{gathered} {(x-y)}^3+z^3 \\ =(x-y+z) \left\{\right(x-y{)}^2-(x-y)z+\left(z{)}^2\right\} \end{gathered}$$

$=(x-y+z)(x^2+y^2-2xy-zx+yz+z^2)$

$64x^3-8y^3$

$=8(8x^3-y^3)$

$=8\left\{\right(2x{)}^3-y^3\}$

$=8(2x-y) (2x{)}^2+2x.y+y^2$

$=8(2x-y)(4x^2+2xy+y^2)$

[এখানে, 40$\times$1 = 40, 10 $\times$4 =40 এবং 10 + 4 = 14]

$x^2+7x-120$

$=x^2+15x-8x-120$

$= x(x + 15) - 8(x + 15)$

= (x + 15)(x - 8) 

[এখানে, (- 120) $\times$ 1 = - 120
15 $\times$ (- 8) = - 120
এবং 15 - 8 = 7 ]

$x^2+7x-120$

$=x^2+15x-8x-120$

= x(x + 15) - 8(x + 15)

= (x + 15)(x - 8) (Ans:)

[এখানে, (- 120)$\times$1 = - 120, 15$\times$(- 8)= -120 এবং 15 - 8 =7]

$x^2+7x-120$

$=x^2+15x-8x-120$

$$\begin{gathered} = x(x + 15) - 8(x + 15) \\ = (x + 15)(x - 8) \end{gathered}$$

[এখানে, 650$\times$ 1 = 650, (- 26)$\times$(- 25) = 650 এবং - 26 - 25 = - 51 ]

$x^2-51x+650$

$=x^2-26x-25x+650$

$= x(x - 26) - 25(x - 26)$

$=(x-26)(x-25)$

[এখানে; (- 80)$\times$1 = - 80, 10$\times$(- 8) = - 80 এবং 10 - 8 = 2 ]

$p^2+2pq-80q^2$

$=p^2+10pq-8pq-80q^2$

$= p(p+10q)-8q(p+10q)$

$=(p+10q)(p-8q)$

[এখানে, -40$\times$1=-40 (- 8)$\times$(5) = - 40 এবং - 8 + 5 = - 3 ]

$x^2-3xy-40y^2$
$=x^2-8xy+5xy-40y^2$

$= x(x-8y)+5y(x-8y)$

$=(x-8y)(x+5y)$

[এখানে, -40$\times$1=-40 (- 8)$\times$(5) = - 40 এবং 8 -5 = 3]

মনে করি, $x^2$-x = a

প্রদত্ত রাশি= $a^2$+ 3a - 40

$=a^2+8a-5a-40$

$= a(a+8)-5(a+8)$

$=(a+8)(a-5)$

$=(x^2-x+8)(x^2-x-5)$ [a এর মান বসিয়ে]

[এখানে, 1$\times$(- 88) = - 88 (- 22) $\times$ 4 = - 88 এবং - 22 + 4 = - 18 ]

ধরি, $a^2+b^2=x$

প্রদত্ত রাশি

$=x^2-18x-88$

$$\begin{gathered} =x^2-22x+4x-88 \\ =x(x-22)+4(x-22 \end{gathered}$$

$=(x-22)(x+4)$

$=(a^2+b^2-22)(a^2+b^2+4)$  [x এর মান বসিয়ে।]

$=(a^2+b^2+4)(a^2+b^2-22)$

[এখানে, - 180 $\times$1 = - 180 , 10(- 18) = - 180 এবং 10 - 18 = - 8 ]

মনে করি,$a^2+7a=x$

প্রদত্ত রাশি $=x^2-8x-180$

$=x^2-18x+10x-180$

$= x(x - 18) + 10(x - 18)$

= (x - 18)(x + 10) [x এর মান বসিয়ে।]

$=(a^2+7a-18)(a^2+7a+10)$

$$\begin{gathered} =(a^2+9a-2a-18)(a^2+5a+2a+10) \\ =\left\{a\right(a+9)-2(a+9\left)\right\}\left\{a\right(a+5)+2(a+5\left)\right\} \end{gathered}$$

$$\begin{gathered} = (a + 9)(a - 2)(a + 5)(a + 2) \\ = (a + 2)(a - 2)(a + 5)(a + 9) \end{gathered}$$

$$\begin{gathered} x^2+(3a+4b)x+(2a^2+5ab+3b^2) \\ =x^2+(3a+4b)x+(2a^2+3ab+2ab+3b^2) \\ =x^2+(3a+4b)x+\left\{a\right(2a+3b)+b(2a+3b\left)\right\} \\ =x^2+\left\{\right(2a+3b)+(a+b\left)\right\}x+(a+b)(2a+3b) \\ =x^2+(3a+4b)x+(a+b)(2a+3b) \\ =x^2+(2a+3b)x+(a+b)x+(a+b)(2a+3b) \\ =x(x+2a+3b)+(a+b)(x+2a+3b) \\ = (x+2a+3b)(x+a+b) \end{gathered}$$

[এখানে, (- 15)$\times$6 = - 90, 9$\times$ (- 10) = - 90 এবং 9 - 10 = - 1 ]

$6x^2-x-15$

$$\begin{gathered} =2x(3x-5)+3(3x-5) \\ =6x^2-10x+9x-15 \end{gathered}$$

$= (3x - 5)(2x + 3)$

$x^2-x-(a+1)(a+2)$

$=x^2-\left\{\right(a+2)-(a+1\left)\right\}x-(a+1)(a+2)$

$=x^2+(a+1)x-(a+2)x-(a+1)(a+2)$

$= x(x + a + 1) - (a + 2)(x + a + 1)$

$= (x + a + 1)(x - a - 2)$

[এখানে, 3$\times$(- 4) = - 12 এখন, 12$\times$(- 1) = - 12 এবং 12 - 1 = 11 ]

$3x^2+11x-4$

$$\begin{gathered} =3x^2+12x-x-4 \\ = 3x(x+4) - (x+4) \\ = (x+4)(3x-1) \end{gathered}$$

[এখানে, 3$\times$(- 12) = - 36 এখন, (- 18)$\times$2 = - 36 এবং - 18 + 2 = - 16]

$3x^2-16x-12$

$=3x^2-18x+2x-12$

$=3x(x-6)+2(x-6)$

$=(x-6)(3x+2)$

এখানে, (-35)$\times$2-70, এখন, (- 14)$\times$ 5 = - 70 এবং 14 + 5 = - 9 ]

$$\begin{gathered} 2x^2-9x-35 \\ =2x^2-14x+5x-35 \\ =2x(x - 7) + 5(x - 7) \\ = (x - 7)(2x + 5) \end{gathered}$$

[এখানে, 2 $\times$2 = 4 এখন, (- 4)$\times$(- 1) = 4 এবং - 4 - 1 = - 5 ]

$2x^2-5xy+2y^2$

$=2x^2-4xy-xy+2y^2$

$= 2x(x - 2y) - y(x - 2y)$

$= (x - 2y)(2x - y)$

$x^3-8(x-y{)}^3$

$=(x{)}^3-\{2(x-y){\}}^3=(x{)}^3-(2x-2y{)}^3$

$=(2y-x)\{x^2+2x^2-2xy+(2x{)}^2-2.2x.2y+\left(2y{)}^2\right\}$

$=(2y-x)(x^7+2x^2-2xy+4x^2-8xy+4y^2)$

$=(2y-x)(7x^2-10xy+4y^2)$

[এখানে, 10 $\times$ (- 6) = - 60
এখন, 15 $\times$ (- 4) = - 60
এবং 15-4=11]

$10p^2 + 11pq-6q^2$

$=10p^2+15pq-4pq -6q^2$

= 5p(2p + 3q) - 2q(2p + 3q)

= (2p + 3q)(5p - 2q) (Ans.)

[এখানে, 2$\times$(-2)=-4,
এখন, (- 4) $\times$ 1 = - 4
এবং - 4 + 1 = - 3 ]

$2 (x+y{)}^2-3(x+y)-2$

$= 2a^2-3a-2$ [x + y -a ধরে ]

$=2a^2-4a+a-2$

= 2a(a - 2) + 1(a - 2)

= (a - 2)(2a + 1)

= (x + y - 2)(2x + 2y + 1) [a এর মান বসিয়ে] (Ans.)

$a{x}^2+(a^2+1) x+a$

$=a{x}^2+a^{2}x+x+a$

= ax(x + a) + 1(x + a)

= (x + a)(ax + 1) (Ans.)

[এখানে, 15 $\times$ (- 12) = - 180
এখন, (- 20) $\times$ 9 = - 180
এবং - 20 + 9 = - 11 ]

$15x^2-11xy-12y^2$

$=15x^2-20xy+9xy-12y^2$

= 5x(3x - 4y) + 3y(3x - 4y)

= (3x - 4y)(5x + 3y) (Ans.)

$a^3-3a^{2}b+3a{b}^2-2b^3$

$=(a^3-3a^{2}b+3a{b}^2-b^3)-b^3$

$=(a-b{)}^3-b^3$

$=(a-b-b) (a-b{)}^2+(a-b).b+(b{)}^2$

$=(a-2b)(a^2-2ab+b^2+ab-b^2+b^2)$

$=(a-2b)(a^2-ab+b^2)$ (Ans.)

$4x^2-y^2=(2x{)}^2-(y{)}^2$

= (2x + y)(2x - y)

Ans. (2x + y)(2x - y)

$6a{b}^2-24a=6a(b^2-4)$

$=6a(b^2-2^2)$

= 6a(b + 2)(b - 2)

Ans. 6a(b + 2)(b - 2)

$x^2+2px+p^2-4$

$=(x^2+2px+p^2)-4$

$=(x+p{)}^2-(2{)}^2$

= (x + p + 2)(x + p - 2)

Ans. (x + p + 2)(x + p - 2)

$x^3+27y^3$

$=(x{)}^3+(3y{)}^3$

$=(x+3y)\{x^2-x.3y+(3y{)}^2\}$

$=(x+3y)(x^2-3xy+9y^2)$

Ans. $(x+3y)(x^2-3xy+9y^2)$

$27a^3-8$

$=(3a{)}^3-(2{)}^3$

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

$=(3a-2)(9a^2+6a+4)$

Ans. $(3a-2)(9a^2+6a+4)$

এখানে, 72 $\times$ 1 = 72
(- 12) $\times$ (- 6) = 72
এবং - 12 - 6 = -18 ]

$x^2-18x+72$

$=x^2-12x-6x+72$

= x(x - 12) - 6(x - 12)

= (x - 12)(x - 6)

Ans. (x - 12)(x - 6)

[এখানে, (- 36) $\times$ 1 = - 36
(- 12) $\times$ 3 = - 36
এবং  - 12 + 3 = - 9 ]

$x^2-9x-36$

$=x^2-12x+3x-36$

= x(x - 12) + 3(x - 12)

= (x - 12)(x + 3)

Ans. (x - 12)(x + 3)

[এখানে, 132 $\times$ 1 = 132
(- 12) $\times$ (- 11) = 132
এবং - 12 - 11 = 23 ]

$x^2-23x+132$

$=x^2-12x-11x+132$

= x(x - 12) - 11(x - 12)

= (x - 12)(x - 11)

Ans. (x - 12)(x - 11)

[ এখানে, 8 $\times$ 9 = 72
(12) $\times$ (6) = 72
এবং 12 + 6 = 18 ]

$8x^2+18x+9$

$=8x^2+12x+6x+9$

$= 4x(2x + 3) + 3(2x + 3)$

= (2x + 3)(4x + 3)

Ans. (2x + 3)(4x + 3)

[ এখানে, 27 $\times$ 2 = 54
9 $\times$ 6 = 54
এবং 9+6=15 ]

$27x^2+15x+2$

$=27x^2+9x+6x+2$

= 9x(3x + 1) + 2(3x + 1)

= (3x + 1)(9x + 2)

Ans. (3x + 1)(9x + 2)

[ এখানে, 1 $\times$ (- 10) = - 10
2 $\times$ (- 5) = - 10
এবং 2-5 = -3 ]

$2a^2-6a-20$

$=2(a^2-3a-10)$

$=2\{a^2-5a+2a-10\}$

= 2{a(a - 5) + 2(a - 5)}

= 2(a - 5)(a + 2)

Ans. 2(a - 5)(a + 2)