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AP EAMCET202319 May 2023Morning ShiftMathematicsQuadratic EquationActual

If , and are the roots of the equation x^3+3 x^2+4 x+5=0 , then the cubic equation whose roots are 1+4 , 1+4 and 1+4 is

Options

  1. Ax^3+9 x^2-21 x+267=0
  2. Bx^3+9 x^2+43 x+267=0
  3. Cx^3+9 x^2+41 x+267=0
  4. Dx^3+9 x^2+43 x+267=0

Correct answer

D. x^3+9 x^2+43 x+267=0

Step-by-step solution

, , are roots of x^3+3 x^2+4 x+5=0 + + =-3 ...(i) + + =4 ...(ii) =-5 ...(iii) aligned & Let A=1+4 , B=1+4 , C=1+4 & then (A+B+C)=(1+4 )+(1+4 )+(1+4 ) & =3+4( + + )=3+4(-3)=-9 & and A B+B C+C A=(1+4 )(1+4 )+(1+4 )(1+4 ) & +(1+4 )(1+4 ) & =1+4 +4 +16 +1+4 +4 +16 +1+4 & +4 +16 & =3+4(2 +2 +2 )+16( + + ) & =3+8( + + )+16( + + ) & =3+8(-3)+16(4)=43 aligned aligned & and A B C=(1+4 )(1+4 )(1+4 ) & =(1+4 +4 +16 )(1+4 ) & =(1+4 +4 +16 +4 +16 +16 +64 ) & =1+4( + + )+16( + + )+64( ) & =1+4(-3)+16(4)+64(-5)=-267 aligned there

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