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Q.If alpha beta gamma are zeroes of 6x³+3x²+5x+1 then find the value of 1/alpha+1/beta+1/gamma. |
| Answer» {tex}\\alpha , \\beta \\text { and } \\gamma{/tex} are zeroes of the polynomial 6x3 + 3x2 - 5x + 1in the given polynomial, 6x3 + 3x2 - 5x + 1a=6, b=3, c=-5, d=1Sum of the roots =\xa0{tex}- \\frac {b}{a}{/tex}{tex}\\alpha + \\beta + \\gamma = - \\frac { 3 } { 6 }{/tex}{tex}\\alpha + \\beta + \\gamma = - \\frac { 1 } { 2 }{/tex}sum of the Product of the roots =\xa0{tex}\\frac {c}{a}{/tex}{tex}\\alpha \\beta + \\beta \\gamma + \\gamma \\alpha = - \\frac { 5 } { 6 }{/tex}Product of the roots =\xa0{tex}- \\frac{d}{a}{/tex}\xa0{tex}\\alpha \\beta \\gamma = - \\frac { 1 } { 6 }{/tex}{tex}\\therefore \\quad \\frac { 1 } { \\alpha } + \\frac { 1 } { \\beta } + \\frac { 1 } { \\gamma } = \\frac { \\alpha \\beta + \\beta \\gamma + \\gamma \\alpha } { \\alpha \\beta \\gamma }{/tex}{tex}= \\frac { - 5 / 6 } { - 1 / 6 } = \\frac { - 5 } { 6 } \\times \\frac { 6 } { - 1 }{/tex}Hence,\xa0{tex}\\alpha ^ { - 1 } + \\beta ^ { - 1 } + \\gamma ^ { -1 } = 5{/tex} | |