Question

Given that 4 is a zero of the polynomial function f left parenthesis x right parenthesis​, find the remaining zeros. f left parenthesis x right parenthesisequalsx cubed minus 6 x squared plus 13 x minus 20 List the remaining zeros​ (other than 4​).

Answer 1

We have been given that 4 is a zero of the polynomial function
`f(x)=x^3-6x^2+13x-20`. We are asked to find the remaining zeros of function.

Since 4 is a zero of f(x), so
`x-4` will be a factor of f(x).

Let us divide our function f(x) by
`x-4`.

`(x^3-6x^2+13x-20)/(x-4)`

`((x-4)(x^2-2x+5))/(x-4)`

Now we will cancel out
`x-4` from numerator and denominator.

`(x^2-2x+5)`

Now we will use quadratic formula to solve for x as:

`(x^2-2x+5)=0`

`x=(-b\pm√(b^2-4ac))/(2a)`

`x=(-(-2)\pm√((-2)^2-4(1)(5)))/(2(1))`

`x=(2\pm√(4-20))/(2)`

`x=(2\pm√(-16))/(2)`

`x=(2\pm√(-1\cdot 16))/(2)`

Now we will use
`i^2=-1`.

`x=(2\pm√(i^2\cdot 16))/(2)`

`x=(2\pm 4i)/(2)`

`x=(2(1\pm 2i))/(2)`

`x=1\pm 2i`

`x=1-2i\text{ or }x= 1+2i`

Therefore, other two zeros of function are
`1-2i\text{ or } 1+2i`.