Find all the zeros of f (x) = x³- 5x²- 7x + 51 given that 4 – i is a zero of f(x).

Step-by-step explanation:

complex roots always occur in pairs.

so two roots are 4-i and 4+i

[x-(4-i)] and [x-(4+i)] are factors of f(x)

or [(x-4)+i] and [(x-4)-i] are factors of f(x)

[(x-4)+i][(x-4)-i]=(x-4)²-i²=x²-8x+16-(-1)=x²-8x+17

x²-8x+17) x³  -5 x²  -7 x +51( x+3

               x³  -8x²  +17x

             -      +        -

                      3x²  -24x +51

                      3x²  -24x +51

                    -        +       -

                      0

so one more factor is x+3

or zero =-3

Hence zeros of f(x) are -3,4-i,4+i

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