কোনো রাশি দুই বা ততোধিক রাশির গুণফলের সমান হলে, শেষোক্ত রাশিগুলোর প্রত্যেকটিকে প্রথমোক্ত রাশির উৎপাদক বা গুণনীয়ক বলা হয়। কোনো বীজগাণিতিক রাশির উৎপাদকগুলো নির্ণয় করার পর রাশিটিকে লব্ধ উৎপাদকগুলোর গুণফলরূপে প্রকাশ করাকে 'উৎপাদকে বিশ্লেষণ বলা হয়।

$p^4+p^2-2=p^4-1+p^2-1$

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

                         $= (p^2-1) (p² + 1) + (p² - 1)$

                            $=(p^2-1)(p^2+1+1)$

                         $=(p^2-1^2)(p^2+2)$

                       $=(p+1)(p-1)(p^2+2)$

$16a^2+\frac{1}{16a^2}-2+16a-\frac{1}{a}$

$=(4a{)}^2-2.4a.\frac{1}{4a}{\left(\frac{1}{4a}\right)}^2+4\left(4a-\frac{1}{4a}\right)$

$={\left(4a-\frac{1}{4a}\right)}^2+4\left(4a-\frac{1}{4a}\right)$

$=\left( 4a - \frac{1}{4a}\right)\left(4a - \frac{1}{4a}+ 4\right)$

প্রদত্ত রাশি= $4P^2-1+2R-R^2$

                 $=4P^2-(R^2-2R+1)$

                  $=4P^2-(R-1{)}^2=(2P{)}^2-(R-1{)}^2$

                  $= (2P + R - 1)(2P - R + 1)$

প্রদত্ত রাশি $=x^2+10x+16-y^2+6y$

                   $=x^2+10x+25-y^2+6y-9$

                     $=x^2+2.x.5+5^2-(y^2-2.y.3+3^2)$

                    $=(x+5{)}^2-(y-3{)}^2$

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

                       $= (x + y + 2) (x - y + 8)$

প্রদত্ত রাশি $=x^4+64=(x^2{)}^2+(8{)}^2$

                                        $=(x^2+8{)}^2-2.x^2.8$

                                          $=(x^2+8{)}^2-16x^2$

                                           $=(x^2+8{)}^2-(4x{)}^2$

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

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

$x^4-3x^2+1=(x^2{)}^2-2.x^2.1+1^2-x^2$

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

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

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

$(2a-b{)}^3-27$

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

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

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

প্রদত্ত রাশি$=b^2-1+2c-c^2$

                  $=b^2-(1-2c+c^2)=b^2-(1-c{)}^2$

                   $=\{b+(l-c\left)\right\} \{b-(l-c\left)\right\}$

                    $= (b + 1 - c) (b - 1 + c)$

                      $=(b - c + 1)(b + c - 1)$

নির্ণেয় উৎপাদকে বিশ্লেষিত রূপ: $(b - c + 1) (b + c - 1)$

$y^4-79y^2+1$

$=(y^2{)}^2+2.y^2.1+1^2-81y^2=(y^2+1{)}^2-(9y{)}^2$

$=(y^2+1+9y)(y^2+1-9y)=(y^2+9y+1)(y^2-9y+1)$

প্রদত্ত রাশি $=2-5x-12x^2$

                   $=2+3x-8x-12x^2$

                    $= 1(2 + 3x) - 4x(2 + 3x) = (2 + 3x)(1 - 4x)$

প্রদত্ত রাশি $= a²b²-8ab-105$

                      $=a^2{b}^2-15ab+ 7ab - 105$

                      $= ab(ab - 15) + 7 (ab - 15)$

                       $= (ab - 15) (ab + 7)$

প্রদত্ত রাশি $=4p^2-4q^2+4q-1$

                  $=4p^2-(4q^2-4q+1)$

                    $=4p^2-\left\{\right(2q{)}^2-2.2q.0.1+1^2\}$

                     $=(2p{)}^2-(2q-1{)}^2$

                     $= (2p + 2q - 1) (2p - 2q + 1)$

নির্ণেয় উৎপাদকে বিশ্লেষিত রূপ: $(2p + 2q - 1 ) (2p - 2q + 1).$

প্রদত্ত রাশি $=49x^2-y^2-8y-16$

                    $=(7x{)}^2-(y^2+8y+16)$

                   $=(7x{)}^2-(y+4{)}^2$

                    $= (7x + y + 4) (7x - y - 4)$

প্রদত্ত রাশি $=a{x}^2+(ab-1)x-b$

                   $=a{x}^2+abx-x-b$

                     $= ax(x + b) - (x + b) = (x + b)(ax - 1)$

প্রদত্ত রাশি$=x^2+4x-21$

                    $=x^2+7x-3x-21$

                   $= x(x + 7) - 3(x + 7) = (x + 7)(x - 3)$

নির্ণেয় উৎপাদকে বিশ্লেষিত রূপ: $(x + 7) (x - 3)$

প্রদত্ত রাশি $=a^4-51a^2+1$

                   $=a^4+2a^2+1-49a^2$

                  $={(a²)}^2-2. a². 1+1^2-{\left(7a\right)}^2$

                    $= (a² - 1)² - \left(7a\right)²= (a²-1+7a) (a²-1-7a)$

                   $=(a^2+7a-1)(a^2-7a-1)$

নির্ণেয় উৎপাদকে বিশ্লেষিত রূপ:  $(a²+7a-1) (a²-7n-1).$

$4x² - 12xy + 9y² - 16z²$

$=(2x{)}^2-2.2x.3y+(3y{)}^2-(4z{)}^2$

$=(2x-3y{)}^2-(4z{)}^2=(2x-3y+4z)(2x-3y-4z).$

প্রদত্ত রাশি $=x^4-38x^2+1$

                     $=(x^2{)}^2-2.x^2.1+(1{)}^2-36x^2$

                       $=(x^2-1{)}^2-(6x{)}^2=(x^2-1+6x) (x^2-1-6x)$

                         $=(x^2+6x-1) (x^2-6x-1).$

$y^2-2yz-4z-4$

$=y^2-4-2yz-4z$

$=y^2-2^2-2yz-4z=(y+2)(y-2)-2z(y+2)$

$= (y + 2)(y - 2 - 2z) = (y + 2)(y - 2z - 2) .$

প্রদত্ত রাশি $=(x+y{)}^3-27y^3$

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

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

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

                      $=(x-2y)(x^2+5xy+13y^2)$

প্রদত্ত রাশি $=x^3+9y^3+(x+y{)}^3$   
                      $=x^3+8y^3+(x+y{)}^3+y^3$

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

                       $=(x+2y)(x²-x.2y+4y²)+(x+y+y)\{(x+y)²-(x+y)y+y²)$

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

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

                           $=(x+2y)(x^2-2xy+4y^2+x^2+xy+y^2)=(x+2y)(2x^2-xy+5y^2)$

প্রদত্ত রাশি $=m^4-7m^2+1$

                  $=(m^2{)}^2+2m^2.1+1^2-9m^2$

                    $=(m^2+1{)}^2-(3m{)}^2$

                       $=(m^2+1+3m)(m^2+1-3m)$

                          $=(m^2+3m+1)(m^2-3m+1).$

$2x^3-3x^2+3x-1$

$= {\left(x\right)}^3 -3.x^2.1.x.1² - {\left(1\right)}^{3 }+ x^3 = {(x - 1)}^3 + \left(x\right)³$

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

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

নির্ণেয় উৎপাদকে বিশ্লেষিত রূপ:  $(2x-1) (x² - x + 1).$

প্রদত্ত রাশি $=a^2-2\left(x+\frac{1}{x}\right)a+4=a^2-2ax-\frac{2a}{x}+4$

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

প্রদত্ত রাশি $=7x^2-x-116$

                    $=7x^2-29x+28x-116$

                     $= x(7x - 29) + 4(7x - 29)$

                         $= (7x - 29)(x + 4)$

প্রদত্ত রাশি$=m^4-7m^2+9$

                  $=(m^2{)}^2-2.m^2.3+(3{)}^2+m^2$

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

                       $=(m^2-3+m)(m^2-3,-m)$

                          $=(m^2+m-3)(m^2-m-3).$

$3x^6-2x^3-1=3x^6-3x^3+x^3-1$

$=3x^3(x^3-1)+1(x^3-1)=(3x^3+1)(x^3-1)$

$=(3x^3+1)(x^3-1^3)=(3x^3+1)(x-1).(x^2+x.1+1^2)$

  $=(3x^3+1)(x-1)(x^2+x+1)=(3x^3+1)(x^2+x+1)(x-1)$

প্রদত্ত রাশি $=a^3-9+(a+1{)}^3$

                   $=a^3-1+(a+1{)}^3-8$

                    $=a^3-(1{)}^3+(a+1{)}^3-(2{)}^3$

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

                      $=(a-1)(a^2+a+1)+(a-1)(a^2+2a+1+2a+2+4)$

                       $=(a-1)(a^2+a+1+a^2+2a+1+2a+2+4)$

                            $=(a-1)(2a^2+5a+8).$

প্রদত্ত রাশি $=x^3-9y^3+(x-y{)}^3$

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

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

                    $=(x-2y)(x^2+x.2y+4y^2)+(x-y-y) \left\{(x-y)²+(x-y).y+y²\right\}$

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

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

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

                         $=(x-2y)(2x^2+xy+5y^2)$

প্রদত্ত রাশি $=a^3+3a^2+3a+2$

                    $=a^3+3a^2.1+3.a.1^2+1+1=(a-1{)}^3+1^3$

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

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

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

প্রদত্ত রাশি $=m^3+m^{-3}+2$

                   $=m^3+\frac{1}{m^3}+2=\frac{(m^3{)}^2+1+2m^3}{m^3}$

                     $=\frac{1}{m^3} \left\{{\left(m^3\right)}^2+2.m^3.1+(1{)}^2\right\}$

                        $=\frac{1}{m^3}\left\{\right(m^3+1{)}^2\}=\frac{1}{m^3}\{(m^3+1^3{)}^2\}$

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

                             $=\frac{1}{m^3}(m+1{)}^2(m^2-m+1{)}^2.$

$a^2-2ab-2b-1$

$=a^2-1^2-2ab-2b=(a+1)(a-1)-2b(a+1)$

$= (a + 1)(a - 1 - 2b) = (a + 1)(a - 2b - 1) .$