If a and b are the zeros of p(x) x2-2x+3 . Find the polynomial whose zeros are a-1/a+1 and b-1/b+1

If a and b are the zeros of p(x) x2-2x+3 . Find the polynomial whose z

1.	If a and b are the zeros of p(x) x2-2x+3 . Find the polynomial whose zeros are a-1/a+1 and b-1/b+1
Answer» Ans. Given: {tex}p\\left( x \\right) = {x^2} - 2x + 3{/tex}\xa0, a and b are zeroes of given polynomial.\xa0We KnowSum of zeroes = {tex}-b\\over a{/tex}=> a + b = 2 andproduct of zeroes = {tex}c\\over a{/tex}=> ab = 3For required polynomial,Sum of zeroes = {tex}{{a - 1} \\over {a + 1}} + {{b - 1} \\over {b + 1}}{/tex}= {tex}{{\\left( {a - 1} \\right)\\left( {b + 1} \\right) + \\left( {b - 1} \\right)\\left( {a + 1} \\right)} \\over {\\left( {a + 1} \\right)\\left( {b + 1} \\right)}}{/tex}= {tex}{{ab + a - b - 1 + ab + b - a - 1} \\over {ab + a + b + 1}}{/tex}\xa0= {tex}{{2ab - 2} \\over {ab + a + b + 1}}{/tex}\xa0= {tex}{{2 \\times 3 - 2} \\over {3 + 2 + 1}}{/tex}\xa0= {tex}{4 \\over 6}{/tex}{tex}\\Rightarrow{/tex}{tex}{{ - b} \\over a} = {4 \\over 6}{/tex}Product of zeroes = {tex}\\left( {{{a - 1} \\over {a + 1}}} \\right)\\left( {{{b - 1} \\over {b + 1}}} \\right){/tex}\xa0= {tex}{{ab - a - b + 1} \\over {ab + a + b + 1}}{/tex}= {tex}{{ab - \\left( {a + b} \\right) + 1} \\over {ab + \\left( {a + b} \\right) + 1}}{/tex}\xa0= {tex}{{3 - 2 + 1} \\over {3 + 2 + 1}}{/tex}{tex}\\Rightarrow{/tex}{tex}{c \\over a} = {2 \\over 6}{/tex}On comparing, we get, a = 6, b = -4\xa0and c = 2Putting these values in the general polynomial {tex}a{x^2} + bx + c{/tex},the required polynomias is\xa0{tex}6x^2-4x+2{/tex}