Question:`A = (x + 1)/(x - 1), B = (x^2 + x)/(x^2 + x - 2), C = x^2/(x^2 + 5x + 6)` এবং `D =(x^2 - y^2)/(x^3 - y^3)` চারটি বীজগাণিতিক ভগ্নাংশ। ক. D কে লঘিষ্ট আকারে প্রকাশ কর। খ.প্রমাণ কর যে, `A -: B xx C = x/(x + 3)` গ. A, B এবং C কে সমহর বিশিষ্ট ভগ্নাংশে প্রকাশ কর।
Answer ক. `D = (x^2 - y^2)/(x^3 - y^3)` ` = ((x + y) (x - y))/((x - y) (x^2 + xy + y^2))` `= ((x + y))/((x^2 + xy + y^2))` উত্তর: ` ((x + y))/((x^2 + xy + y^2))` খ. দেওয়া আছে, `A = (x + 1)/(x - 1), B = (x^2 + x)/(x^2 + x - 2),`এবং ` C = x^2/(x^2 + 5x + 6)` `:. A -: B xx C = (x + 1)/(x - 1) -: (x^2 + x)/(x^2 + x - 2) xx x^2/(x^2 + 5x + 6)` `= (x + 1)/(x - 1) xx (x^2 + x - 2)/(x^2 + x) xx x^2/(x^2 + 5x + 6)` `= (x + 1)/(x - 1) xx (x^2 + 2x - x - 2)/(x(x + 1)) xx x^2/(x^2 + 3x + 2x + 6)` `= (x + 1)/(x - 1) xx ((x + 2)(x - 1))/(x(x + 1)) xx x^2/((x + 3)(x + 2))` `= x/(x + 3)` `:. A -: B xx C = x/(x + 3)` (প্রমাণিত) গ. `A = (x + 1)/(x - 1), B = (x^2 + x)/(x^2 + x - 2)`এবং `C = x^2/(x^2 + 5x + 6)` এখানে A এর হর x = 1 B এর হর `= x^2 + x - 2` `= x^2 + 2x - x - 2` `= x(x + 2) - 1(x + 2)` `= (x + 2) (x - 1)` C এর হর `= x^2 + 5x + 6` `= x^2 + 3x + 2x + 6` `= x(x + 3) + 2(x + 3)` `= (x + 3) (x + 2)` :. A, B, C এর হরগুলোর ল,সা,গু = (x - 1) (x + 2) (x + 3) প্রাপ্ত ল,সা,গু কে প্রদত্ত ভগ্নাংশগুলির হর দ্বারা ভাগ করলে ভাগফল হয় যথাক্রমে (x + 2) (x + 3), (x + 3) ও (x - 1) :. `A = (x + 1)/(x - 1) = ((x + 1)(x + 2)(x + 3))/((x - 1)(x + 2)(x + 3))` `B = (x^2 + x)/(x^2 + x - 2) = ((x^2 + x) xx (x + 3))/((x + 2)(x - 1) xx (x + 3)) = ((x^2 + x)(x + 3))/((x - 1)(x + 2)(x + 3))` এবং `C = x^2/(x^2 + 5x + 6), = (x^2 xx (x - 1))/((x + 2)(x + 3) xx (x - 1)) = (x^2 - (x - 1))/((x - 1)(x + 2)(x + 3))` :. নির্ণেয় ভগ্নাংশগুলি `((x + 1)(x + 2)(x + 3))/((x - 1)(x + 2)(x + 3))` `((x^2 + x)(x + 3))/((x - 1)(x + 2)(x + 3)), (x^2 - (x - 1))/((x - 1)(x + 2)(x + 3))` (Ans)
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`A = (x + 1)/(x - 1), B = (x^2 + x)/(x^2 + x - 2), C = x^2/(x^2 + 5x + 6)` abong `D =(x^2 - y^2)/(x^3 - y^3)` charoti bijoganitik bhgnangsho. ka. D ke lghisht akare prokasho karo. kh.proman kar je, `A -: B xx C = x/(x + 3)` ga. A, B abong C ke shomohor bishisht bhgnangshe prokasho karo.