# Find all complex zeros of function, given one of the zeros is 4+i. Then write the linear factorization of the function: g(x) = 4x^5 - 57x^4 + 287x^3 - 547x^2 + 83x + 510

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The function `g(x)=4x^5 - 57x^4 + 287x^3 - 547x^2 + 83x + 510` .

One of the zeros is 4 + i, there is another complex zero equal to 4 - i. To write the linear factorization the other 3 roots have to be determined.

`4x^5 - 57x^4 + 287x^3 - 547x^2 + 83x + 510`

=> 4x^5 - 20x^4 - 37x^4 + 185x^3 + 102x^3 - 510x^2 - 37x^2 + 185x - 102x + 510

=> 4x^4(x - 5) - 37x^3(x - 5) + 102x^2(x - 5) - 37x(x - 5) - 102(x - 5) = 0

=> (x - 5)(4x^4 - 37x^3 + 102x^2 - 37x - 102) = 0

=> (x - 5)(4x^4 - 8x^3 - 29x^3 + 58x^2 + 44x^2 - 88x + 51x - 102) = 0

=> (x - 5)(4x^3(x - 2) - 29x^2(x - 2) + 44x(x - 2) + 51(x - 2)) = 0

=> (x - 5)(x - 2)(4x^3 - 29x^2 + 44x + 51) = 0

=> (x - 5)(x - 2)(4x^3 + 3x^2 - 32x^2 - 24x + 68x + 51) = 0

=> (x - 5)(x - 2)(x^2(4x + 3) - 8x(4x + 3) + 17(4x + 3)) = 0

=> (x - 5)(x - 2)(4x + 3)(x^2 - 8x + 17) = 0

x - 5 = 0

=> x = 5

x - 2 = 0

=> x = 2

4x + 3 = 0

=> x = -3/4

The other 2 roots are given as 4 + i and 4 - i

**This gives the roots as `{4 + i, 4 - i, -3/4, 2, 5}` . The linear factorization is **`g(x) = (x - 2)(x - 5)(4x + 3)(x - 4 - i)(x - 4 + i) `