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Let-x-1-x-2-x-3-be-the-roots-of-the-equation-x-3-3x-5-0-Then-the-value-of-expression-x-1-1-x-1-x-2-1-x-2-x-3-1-x-3-is-equal-to-

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Question Number 162367 by cortano last updated on 29/Dec/21
Let x_1 ,x_2 ,x_3 be the roots of the equation x^3 +3x+5=0 . Then the value of expression (x_1 +(1/x_1 ))(x_2 +(1/x_2 ))(x_3 +(1/x_3 )) is equal to
Letx1,x2,x3betherootsoftheequationx3+3x+5=0.Thenthevalueofexpression(x1+1x1)(x2+1x2)(x3+1x3)isequalto
Answered by mr W last updated on 29/Dec/21
x_1 +x_2 +x_3 =0 x_1 x_2 +x_2 x_3 +x_3 x_1 =3 x_1 x_2 x_3 =−5 (x_1 +(1/x_1 ))(x_2 +(1/x_2 ))(x_3 +(1/x_3 )) =(((x_1 ^2 +1)(x_2 ^2 +1)(x_3 ^2 +1))/(x_1 x_2 x_3 )) =(((x_1 x_2 x_3 )^2 +(x_1 ^2 x_2 ^2 +x_2 ^2 x_3 ^2 +x_3 ^2 x_1 ^2 )+(x_1 ^2 +x_2 ^2 +x_3 ^2 )+1)/(x_1 x_2 x_3 )) =(((x_1 x_2 x_3 )^2 +(x_1 x_2 +x_2 x_3 +x_3 x_1 )^2 −2x_1 x_2 x_3 (x_1 +x_2 +x_3 )+(x_1 +x_2 +x_3 )^2 −2(x_1 x_2 +x_2 x_3 +x_3 x_1 )+1)/(x_1 x_2 x_3 )) =(((−5)^2 +(3)^2 −2(−5)(0)+(0)^2 −2(3)+1)/(−5)) =((25+9−6+1)/(−5)) =−((29)/5)
x1+x2+x3=0x1x2+x2x3+x3x1=3x1x2x3=−5(x1+1x1)(x2+1x2)(x3+1x3)=(x12+1)(x22+1)(x32+1)x1x2x3=(x1x2x3)2+(x12x22+x22x32+x32x12)+(x12+x22+x32)+1x1x2x3=(x1x2x3)2+(x1x2+x2x3+x3x1)2−2x1x2x3(x1+x2+x3)+(x1+x2+x3)2−2(x1x2+x2x3+x3x1)+1x1x2x3=(−5)2+(3)2−2(−5)(0)+(0)2−2(3)+1−5=25+9−6+1−5=−295

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