Find a cubic polynomial whose zeroes are 3, 5 and -2.

1.	Find a cubic polynomial whose zeroes are 3, 5 and -2.
Answer» Let\xa0{tex}\\alpha,\\mathrm\\beta\\;\\mathrm{and}\\;\\mathrm\\gamma{/tex}\xa0be the zeroes of the given polynomial.\xa0Then, we have\xa0{tex}\\alpha{/tex}\xa0= 3,\xa0{tex}\\beta{/tex}\xa0= 5 and\xa0{tex}\\gamma{/tex}\xa0= -2\xa0Hence{tex}\\alpha + \\beta + \\gamma{/tex}\xa0= 3 + 5 - 2 = 6 ...............(1){tex}\\alpha \\beta + \\beta \\gamma + \\gamma \\alpha{/tex}\xa0= 3(5) + 5(-2) + (-2)3 = 15 - 10 - 6 = -1 ................(2){tex}\\alpha \\beta \\gamma{/tex}\xa0= 3(5)(-2) = -30 .............(3)Now, a cubic polynomial whose zeros are {tex}\\alpha , \\beta{/tex}\xa0and\xa0{tex}\\mathrm\\gamma{/tex}\xa0is equal top(x) = x3\xa0-\xa0{tex}( \\alpha + \\beta + \\gamma ) x ^ { 2 } + ( \\alpha \\beta + \\beta y + \\gamma \\alpha ) x - \\alpha \\beta \\gamma{/tex}On substituting values from (1),(2) and (3) we get{tex}\\mathrm p(\\mathrm x)=\\mathrm x^3-(6)\\mathrm x^2+(-1)\\mathrm x-(-30){/tex}= x3\xa0- 6x2 - x + 30 3x^3-5x^2-2x

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