To find the roots of the given polynomial, here are the steps:
1. Equate the function to zero.
[tex]x^3+6x^2+12x+7=0[/tex]
2. Find a value of x that would make the equation zero. By trial and error, we can use x = -1. Let's check if this makes the equation zero. Let's replace x with -1 in the equation above.
[tex]\begin{gathered} (-1)^3+6(-1)^2+12(-1)+7=0 \\ -1+6-12+7=0 \\ 0=0 \end{gathered}[/tex]
Since the equation is equal to zero when x = -1, we can say that (x + 1) is a factor of the polynomial.
3. Let's find the other factor. To get the other factor of the polynomial, let's divide the polynomial by (x + 1) using synthetic division.
Hence, the other factor of the polynomial is (x² + 5x + 7).
4. Equate the other factor to zero and find the value of x. Use quadratic formula.
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]
In the second factor, a = 1, b = 5, and c = 7. Let's plug this in to the formula above.
[tex]x=\frac{-5\pm\sqrt[]{5^2-4(1)(7)}}{2(1)}[/tex]
Simplify.
[tex]\begin{gathered} x=\frac{-5\pm\sqrt[]{25-28}}{2} \\ x=\frac{-5\pm\sqrt[]{-3}}{2} \\ x=\frac{-5\pm i\sqrt[]{3}}{2} \end{gathered}[/tex]
Therefore, the roots of the given polynomial are x = -1 and x = (-5 ± i√3)/2 as shown above.
Based on the options, only Option B is a root of the given polynomial.
