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 $x-4$.

$\frac{x^3-6x^2+13x-20}{x-4}$

$\frac{(x-4)(x^2-2x+5)}{x-4}$

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:

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

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$x=\frac{-(-2)\pm\sqrt{(-2)^2-4(1)(5)}}{2(1)}$

$x=\frac{2\pm\sqrt{4-20}}{2}$

$x=\frac{2\pm\sqrt{-16}}{2}$

$x=\frac{2\pm\sqrt{-1\cdot 16}}{2}$

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

$x=\frac{2\pm\sqrt{i^2\cdot 16}}{2}$

$x=\frac{2\pm 4i}{2}$

$x=\frac{2(1\pm 2i)}{2}$

$x=1\pm 2i$

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

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