Need help asap thank you

Question

Find the completely factored form and the polynomial form.

degree: 4

zeros: 1,-4,3i

solution point: f(0)=-6

Answer 1

zeroes 1, -4, 3i

(x-3i)(x+3i)(x-1)(x+4) is in factored form with the required zeros

or if you want real factors: (x^2+9)(x-1)(x+4)

but the constant term will be 9(-1)(4) = -36. you want -6

so stick a coefficent 1/6 in front of the factors to get

(1/6)(x^+9)(x-1)(x+4) or (1/6)(x-3i)(x+3i)(x-1)(x+4)

or in 4th degree polynomial form multiply it out

x^4/6 + x^3/2 + 5x^2/6 + 9x/2 - 6

Answer 2

If 3i is a root, then so is negative 3i. The polynomial will be of the form a(x-1)(x+4)(x-3i)(x+3i) = a(x^4+3x^3+5x^2+27x-36). f(0) = -36a, so a = 1/6. So the final answer is 1/6x^4 + 1/2x^3 + 5/36x^2 + 9/2x - 6.