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Obtain all zeroes of 3x⁴-15x³+13x²+25x-30 if two of it\'s zero\'s are √5/3 and -√5/3 |
| Answer» We have the polynomial f(x) = (3x4 - 15x3 + 13x2 + 25x - 30).Since\xa0{tex}\\sqrt { \\frac { 5 } { 3 } } \\text { and } - \\sqrt { \\frac { 5 } { 3 } }{/tex}\xa0are the zeros of f(x),i.e.,\xa0{tex}x=-\\sqrt{\\frac53}{/tex}\xa0and {tex}x=\\sqrt{\\frac53}{/tex}i.e.,\xa0{tex}x+\\sqrt{\\frac53}=0\\;and\\;x-\\sqrt{\\frac53}=0{/tex} so it follows that eachone of\xa0{tex}\\left( x - \\sqrt { \\frac { 5 } { 3 } } \\right) \\operatorname { and } \\left( x + \\sqrt { \\frac { 5 } { 3 } } \\right){/tex}\xa0is a factor of f(x).so using\xa0{tex}\u200b\u200b\\left(a+b\\right)\\left(a-b\\right)=a^2-b^2{/tex}, we get{tex}\\therefore \\quad \\left( x - \\frac { \\sqrt { 5 } } { \\sqrt { 3 } } \\right) \\left( x + \\frac { \\sqrt { 5 } } { \\sqrt { 3 } } \\right) = 0 \\Rightarrow \\left( x ^ { 2 } - \\frac { 5 } { 3 } \\right)= 0 \\Rightarrow \\frac { \\left( 3 x ^ { 2 } - 5 \\right) } { 3 }= 0{/tex}\xa0is also a factor of f(x).Consequently, (3x2\xa0- 5) is a factor of f{x).On dividing the polynomial by (3x2 - 5), we get{tex}\\therefore{/tex}\xa0f(x) = 3x4 - 15x3 + 13x2 + 25x - 30= (3x2\xa0- 5 )(x2\xa0- 5x + 6).By middle term factorisation. We get,={tex}\\left(3x^2-5\\right)\\left(x^2-2x-3x+6\\right){/tex}=({tex}\\lbrack(\\sqrt{3x})^2-\\left(\\sqrt5\\right)^2\\rbrack{/tex}{tex}\\left[x\\left(x-2\\right)-3\\left(x-2\\right)\\right]{/tex}By using\xa0{tex}a^2-b^2=\\left(a+b\\right)\\left(a-b\\right){/tex}we get,{tex}= ( \\sqrt { 3 } x + \\sqrt { 5 } ) ( \\sqrt { 3 } x - \\sqrt { 5 } ) ( x - 2 ) ( x - 3 ){/tex}{tex}\\therefore{/tex}\xa0f(x) = 0 , so either factors can equated to zero to get the roots\xa0{tex}\\Rightarrow ( \\sqrt { 3 } x + \\sqrt { 5 } ) = 0 \\text { or } ( \\sqrt { 3 } x - \\sqrt { 5 } ) = 0{/tex}{tex}\\Rightarrow{/tex}(x - 2) = 0 or (x - 3) = 0{tex}\\Rightarrow x = - \\sqrt { \\frac { 5 } { 3 } } \\text { or } x = \\sqrt { \\frac { 5 } { 3 } }{/tex}\xa0or x = 2 or x = 3Hence, we get all zeros of f(x) are\xa0{tex}\\sqrt { \\frac { 5 } { 3 } } , - \\sqrt { \\frac { 5 } { 3 } }{/tex}, 2 and 3. | |