Question

Can you help me with #7? X^3-2x^2+3x-6 = 0Please follow prompt b

Answer 1

Given:

The polynomial is given as,

`x^3-2x^2+3x-6=0`

The objective is to factor the polynomial completely.

Step-by-step explanation:

Consider x = 2 in the given equation.

`\begin{gathered} f(2)=2^3-2(2)^2+3(2)-6 \\ =8-8+6-6 \\ =0 \end{gathered}`

Thus, (x -2) is a factor of the polynomial.

Now, using synthetic division,

Thus, the polynomial equation will be,

`x^2+3=0\text{ . . . . .(1)}`

On factorizing the equation (1),

`\begin{gathered} x^2=-3 \\ x=\pm\sqrt[]{-3} \\ x=\pm i\sqrt[]{3} \\ x=i\sqrt[]{3},-i\sqrt[]{3} \end{gathered}`

Hence, the factors of the polynomial are (x-2), (x+i√3), (x-i√3).