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| 1. |
Verify that 3,-1,-1/3,are the zeroes of a cubic polynomial p(x) =3x²- 5x²-11x²-3 |
| Answer» Given polynomial is p(x) = 3x3 -5x2 - 11x - 33Put x = 3 in p(x), we get{tex} \\therefore{/tex}p(3) = 3 {tex} \\times{/tex}\xa033 - 5 {tex} \\times{/tex}\xa032 - 11 {tex} \\times{/tex}\xa03 - 3 = 81 - 45 - 33 - 3 = 0Put x = -1 in p(x), we getp(-1) = 3 {tex} \\times{/tex}\xa0(-1)3 - 5\xa0{tex} \\times{/tex} (-1)2 -11 {tex} \\times{/tex}\xa0(-1) - 3 = - 3 - 5 + 11 - 3 = 0Put x = {tex}- \\frac { 1 } { 3 }{/tex} in p(x), we getand,\xa0{tex} p \\left( - \\frac { 1 } { 3 } \\right){/tex}={tex} 3 \\times \\left( - \\frac { 1 } { 3 } \\right) ^ { 3 } - 5 \\times \\left( - \\frac { 1 } { 3 } \\right) ^ { 2 } - 11 \\times \\left( - \\frac { 1 } { 3 } \\right) - 3{/tex}={tex} - \\frac { 1 } { 9 } - \\frac { 5 } { 9 } + \\frac { 11 } { 3 } - 3{/tex}= 0\xa0So, 3,-1 and\xa0{tex} -\\frac 13{/tex}\xa0are the zeros of polynomial p(x).Let,\xa0{tex} \\alpha = 3 , \\beta = - 1 \\text { and } \\gamma = - \\frac { 1 } { 3 }{/tex} .Then,{tex} \\alpha + \\beta + \\gamma{/tex} ={tex} 3 - 1 - \\frac { 1 } { 3 } = \\frac { 5 } { 3 }{/tex}, and\xa0{tex} - \\frac { \\text { Coefficient of } x ^ { 2 } } { \\text { Coefficient of } x ^ { 3 } } = - \\left( \\frac { - 5 } { 3 } \\right) = \\frac { 5 } { 3 }{/tex}{tex} \\therefore \\quad \\alpha + \\beta + \\gamma = - \\frac { \\text { Coefficient of } x ^ { 2 } } { \\text { Coefficient of } x ^ { 3 } }{/tex}{tex} \\alpha \\beta + \\beta \\gamma + \\gamma \\alpha{/tex} = {tex} 3 \\times ( - 1 ) + ( - 1 ) \\times \\left( - \\frac { 1 } { 3 } \\right) + \\left( - \\frac { 1 } { 3 } \\right) \\times 3{/tex}=\xa0{tex} - 3 + \\frac { 1 } { 3 } - 1 = - \\frac { 11 } { 3 }{/tex}and,\xa0{tex} \\frac { \\text { Coefficient of } x } { \\text { Coefficient of } x ^ { 3 } } = - \\frac { 11 } { 3 }{/tex}{tex} \\therefore \\quad \\alpha \\beta + \\beta \\gamma + \\gamma \\alpha = \\frac { \\text { Coefficient of } x } { \\text { Coefficient of } x ^ { 3 } }{/tex}{tex} \\alpha \\beta \\gamma = 3 \\times ( - 1 ) \\times \\left( - \\frac { 1 } { 3 } \\right) = 1{/tex}......... (i)And,\xa0{tex} - \\frac { \\text { Constant term } } { \\text { Coefficient of } x ^ { 3 } } = - \\left( \\frac { - 3 } { 3 } \\right) = 1{/tex} ........ (ii)From (i) and (ii){tex} \\therefore \\quad \\alpha \\beta \\gamma = - \\frac { \\text { Constant term } } { \\text { Coefficient of } x ^ { 3 } }{/tex} | |