Question

Given that 4 is a zero of the polynomial function f(x), find the remaining zeros.f(x) = x³ - 6x² + 25x - 68List the remaining zeros (other than 4).4(Simplify your answer. Type an exact answer, using radicals and i as needed. Use a cc

Answer 1

`\begin{gathered} x=1+4i \\ x=1-4i \end{gathered}`

Step-by-step explanation

Given:

`\begin{gathered} f(x)=x^3-6x^2+25x-68 \\ \end{gathered}`

Also,

One of the zeros: x = 4

Desired Outcome:

List the remaining zeros using radicals and i.

Simplify the polynomial using x - 4 = 0

Determine the remaining polynomials by simplifying x^2 - 2x + 17 = 0 using the quadratic formula

`x=(-b\pm√(b^2-4ac))/(2a)`

where:

a = 1,

b = -2

c = 17

Substitute the values

`\begin{gathered} x=(-(-2)\pm√((-2)^2-4(1)(17)))/(2(1)) \\ x=(2\pm√(4-68))/(2) \\ x=(2\pm√(-64))/(2) \\ x=(2\pm√(64*-1))/(2) \\ x=(2\pm(√(64)*√(-1)))/(2) \\ x=(2\pm8√(-1))/(2) \\ x=1\pm4√(-1) \\ \text{ Recall: }√(-1)\text{ = }i \\ x=1\pm4i \\ x=1+4i\text{ }or \\ x=1-4i \end{gathered}`