Let f(x)=x^4+a x^3+b x^2+c be a polynomial with real coefficients such that f(1)=-9 . Suppose that i 3 is a root of the equation 4 x^3+3 a x^2+2 b x=0 , where i= -1 . If ₁, ₂, ₃ , and ₄ are all the roots of the equation f(x)=0 , then | ₁ |^2+ | ₂ |^2+ | ₃ |^2+ | ₄ |^2 is equal to ________.

Step-by-step solution

f(1)=1+a+b+c=-9 a+b+c=-10.....(1)4 x^3+3 a x^2+2 b x=0 roots are 3 i,- 3 i, 0 4 x^2+3 a x+2 b=0 < ^ 3i _ - 3i a =0 ~ &~ 2 ~b 4 =( 3 i )(- 3 i ) b =6 use a , b in (1) c =-16 array ll & f ( x )= x ^4+6 x ^2-16=0 & ( x ^2+8 ) ( x ^2-2 )=0 & x = 8 i , 2 array | ₁ |^2+ | ₂ |^2+ | ₃ |^2+ | ₄ |^2=20