যদি হয় a b c 0 তবে দেখাও যে ক, বীজগণিতিক রাশি, গণিত

Question:যদি হয়, a + b + c = 0 তবে দেখাও যে,

 ক.` a^3 + b^3 + c^3 = 3abc`

 খ.`(b + c)^2/(3ab) + (c + a)^2/(3ca) + (a + b)^2/(3ab) = 1`

Answer ক. দেওয়া আছে, a + b + c = 0

        বা, a + b = c

        বা, `(a + b)^3 = (-c)^3` [উভয়পক্ষকে ঘন করে]

        বা, `a^3 + b^3 + 3ab(a + b) = -c^3`

        বা, `a^3 + b^3 + 3ab(-c) = -c^3` [:. a + b = - c]

          `:. a^3 + b^3 + c^3 = 3abc`  (দেখানো হলো)

  
  খ. দেওয়া আছে, 
           a + b + c = 0
   
       :. a + b = -c, b + c = - a

          c + a = -b

   বামপক্ষ   `= (b + c)^2/(3bc) + (c + a)^2/(3ca) + (a + b)^2/(3ab)`

              `= (-a)^2/(3bc) + (-b)^2/(3ca) + (-c)^2/(3ab)`

              `= a^2/(3bc) + b^2/(3ca) + c^2/(3ab)`

              `= (a^3 + b^3 + c^3)/(3abc)`

             `= ((a + b)^3 - 3ab(a + b) + c^3)/(3abc)` [সূত্র ব্যবহার করে]

             `= ((-c)^3 - 3ab(-c) + c^3)/(3abc)` [:. a + b = - c]

            `= (-c^3 + 3abc + c^3)/(3abc)`

            `= (3abc)/(3abc)`

            `= 1`

              = ডানপক্ষ

    `:. (b + c)^2/(3bc) + (c + a)^2/(3ca) + (a + b)^2/(3ab) `

      `= 1`  (দেখানো হলো)

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jodi hoyo, a + b + c = 0 tobe dekhao je, ka.` a^3 + b^3 + c^3 = 3abc` kh.`(b + c)^2/(3ab) + (c + a)^2/(3ca) + (a + b)^2/(3ab) = 1`

Question:যদি হয়, a + b + c = 0 তবে দেখাও যে, ক.` a^3 + b^3 + c^3 = 3abc` খ.`(b + c)^2/(3ab) + (c + a)^2/(3ca) + (a + b)^2/(3ab) = 1`