91Ó°ÊÓ

Write down the coefficients of the original polynomial, \(x^{3}-4 x^{2}+x+6\), those are \[1,-4,1,6\]. Set up synthetic division with these coefficients and -1 below the bar (since we are dividing by \(x+1\), we use \(-1\). Write \(-1\) in the left column and then proceed with the synthetic division: 1. Bring down the first coefficient 1. 2. Multiply \(-1\) (the number in our left column) by this coefficient and write the result under the next coefficient and add them up. This gives \[-1 \times 1 = -1\]. Write this beneath \(-4\) and add to get \(-5\). 3. Repeat this process for all remaining coefficients. Finally, the row above the bar represents the original polynomial and the row under the under the bar represents the coefficients of the result of the division.

The resulting coefficients obtained after synthetic division are \(1, -5, -4\). These coefficients must be used to form the resulting polynomial of the synthetic division. The degree of this polynomial would be one less than the original polynomial (cubic), so it will be quadratic. Hence, the resulting quadratic is given by \(x^{2} - 5x - 4\).

Now, solve the resulting polynomial \(x^{2} - 5x - 4 = 0\) for its roots. This is a standard quadratic equation that should be solvable using the quadratic formula, \(x=[-b\pm\sqrt{b^{2}-4ac}]/[2a]\). Substituting the values \(a=1, b=-5, c=-4\) into this formula will result in two solutions, which represent the other two zeros of the original function.