$6x-9-4x^2$ এবং $4x^2+6x+9$ দুইটি বীজগণিতীয় রাশি।

$A+B=x^3-x^2{y}^2+y^3+x^3+x^2{y}^2+y^3$

           $=2x^3+2y^3=2(x^3+y^3)$ (Ans.)

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

$\therefore$ $AB=x^6+2x^3{y}^3-x^4{y}^4+y^6(Ans.)$

C = 7a + (- 5b) - [2a + (3c - 8b) + {3a - (5b + 9c)}]

= 7a - 5b - [2a + 3c - 8b + {3a - 5b - 9c}]

= 7a - 5b - [2a + 3c - 8b + 3a - 5b - 9c]

= 7a - 5b - [5a - 13b - 6c] = 7a - 5b - 5a + 13b + 6c

$\therefore$ C = 2a + 8b + 6c (প্রমাণিত)

দেওয়া আছে, $Q=4x^2+9y^2-6xy$

দেওয়া আছে,

$P=16x^4+81y^4+36x^2{y}^2$ এবং  $Q=4x^2+9y^2-6xy$

$\therefore$ নির্ণেয় ভাগফল $4x^2+9y^2+6xy$ (Ans.)

দেওয়া আছে,  R = [8b - 3{2a + 3(2b + 5) - 5(b + 3)}] - 3b

বামপক্ষ, R + 6a

= [8b - 3{2a + 3(2b + 5) - 5(b, 3)}] - 3b + 6a

= [8b - 3{2a + 6b + 15-5b - 15}] - 3b + 6a

= [8b - 3{2a + b}] - 3b + 6a = [8b - 6a - 3b] - 3b + 6a

= 5b - 6a - 3b + 6a = 2b = ডানপক্ষ

$\therefore$ R + 6a = 2b (দেখানো হলো)