If the zeros of the polynomial X³-3x²+x+1are a-b, a+B, find a and b

1.	If the zeros of the polynomial X³-3x²+x+1are a-b, a+B, find a and b
Answer» Given polynomial is f(x) = x3\xa0- 3x2\xa0+ x + 1Let\xa0{tex} \\alpha{/tex}\xa0= (a - b),\xa0{tex} \\beta{/tex}\xa0= a and\xa0{tex} \\gamma{/tex}\xa0= (a + b)Now,\xa0{tex} \\alpha + \\beta + \\gamma{/tex}\xa0=\xa0{tex} - \\frac { ( - 3 ) } { 1 }{/tex}⇒\xa0(a - b) + a + ( a + b ) = 3⇒ a - b + a + a+ b = 3⇒ a + a + a = 3⇒\xa03a = 3⇒ a = 3/3⇒\xa0a = 1Also,\xa0{tex} \\alpha \\beta + \\beta y + \\gamma \\alpha = \\frac { 1 } { 1 }{/tex}⇒\xa0(a - b)a + a (a + b) + (a + b)(a - b) = 1\xa0⇒\xa0a2\xa0- ab + a2\xa0+ab + a2\xa0- b2\xa0= 1⇒\xa03a2\xa0- b2\xa0= 1 ( ∵ a = 1)⇒\xa03(1)2\xa0- b2\xa0= 1( ∵ a = 1)⇒ 3 - b2 = 1⇒\xa0b2\xa0= 2⇒\xa0b =\xa0{tex} \\pm \\sqrt{2}{/tex}Hence, a = 1 and b =\xa0{tex} \\pm \\sqrt{2}{/tex}

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