Question

Could you please tell me the answers and explain how to find them?

Answer 1

`x=2,x=3+i√(2),x=3-i√(2)`

Explanation:

Given the function:

`y=x^3-8x^2+23x-22`

As seen on the graph, from the point (2,0) one of the roots of the function:

`x=2`

This means that x-2 is a factor of the polynomial y.

To get the other roots, first, divide y by x-2:

So, we have that:

`\begin{gathered} y=x^(3)-8x^(2)+23x-22 \\ =(x-2)(x^2-6x+11) \end{gathered}`

We then solve the quadratic equation for the other roots of y:

`\begin{gathered} x^2-6x+11=0 \\ x=(-b\pm√(b^2-4ac) )/(2a),a=1,b=-6,c=11 \\ \\ x=(-(-6)\pm√((-6)^2-4(1)(11)))/(2*1) \\ =(6\pm√(36-44))/(2) \\ =(6\pm√(-8))/(2) \\ =(6)/(2)\pm(i√(8))/(2) \\ =3\pm(i2√(2))/(2) \\ x=3\pm i√(2) \end{gathered}`

The other two roots of y are:

`\begin{gathered} x=3+i√(2) \\ \begin{equation*} x=3-\mathrm{i}√(2) \end{equation*} \end{gathered}`

Thus, the exact roots of y are:

`\begin{gathered} x=3+i√(2) \\ \begin{equation*} x=3-\mathrm{i}√(2) \end{equation*} \\ x=2 \end{gathered}`