Math, asked by Aswathiachu, 11 months ago

find the zeros of the polynomial x square +1/6 x -2 and verify the relationship between the zeros and coefficient ​

Answers

Answered by BHL
2
\sf{f(x) = x^2 + {\dfrac{1}{6}} x - 2}


\sf{f(x) = {\dfrac{6x^2 + x - 12}{6}} }


\sf{To \ find \ zeroes, \ f(x) = 0}


\sf{\therefore \ {\dfrac{6x^2 + x - 12}{6}} = 0}


\sf{6x^2 + x - 12 = 0}


\sf{\underline{By \ Middle \ Term \ Factorisation}}


\sf{6x^2 + 9x - 8x - 12 = 0}


\sf{3x(2x + 3) - 4(2x + 3) = 0}


\sf{(3x - 4)(2x + 3) = 0}


\sf{\underline{Using \ Zero \ Product \ Rule}}


\sf{3x - 4 = 0 \ and \ 2x + 3 = 0}


\sf{x = {\dfrac{4}{3}} \ and \ x = {\dfrac{- 3}{2}} }


\sf{Let \ \alpha \ and \ \beta \ be \ the \ zeroes.}


\sf{\therefore \ \alpha = {\dfrac{4}{3}}, \ \beta = {\dfrac{- 3}{2}} }


\large\sf\red{\underline{\underline{Verification \ -}}}


\sf{On \ comparing \ the \ above \ polynomial \ with}

\sf{ax^2 + bx + c = 0, \ we \ get}


\sf{a = 1, \ b = {\dfrac{1}{6}}, \ c = - 2}


\sf{Now,}


\sf{\underline{Sum \ of \ zeroes \ :}}


\sf{\alpha + \beta = {\dfrac{- b}{a}}}


\sf{ {\dfrac{4}{3}} + {\dfrac{- 3}{2}} = - {\dfrac{ {\dfrac{1}{6}} }{1}}}


\sf{- {\dfrac{1}{6}} = - {\dfrac{1}{6}} }


\sf{\underline{Product \ of \ zeroes \ :}}


\sf{\alpha \beta = {\dfrac{c}{a}} }


\sf{ {\dfrac{4}{3}} * {\dfrac{- 3}{2}} = {\dfrac{- 2}{1}} }


\sf{ - 2 = - 2}
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