Find the bounds on the real zeros for `f(x)=x^4+6x^3-2x-4`
We use synthetic division. After using synthetic division we are left with a quotient and remainder; the last number in the process is the remainder, and the rest of the numbers are the coefficients of the quotient.
If the coefficients alternate from nonpositive to nonnegative or vice versa, the divisor is a lower bound on the real roots. If all the coefficients and remainder are positive, the divisor is an upper bound on the real roots.
(1) Looking at a graph or table we suppose that -6 is a lower bound on the real roots. Applying synthetic division we get 1 0 0 -2 8 as the result; nonneg,nonpos,nonneg,nonpos,nonneg. So -6 is a lower bound on the real roots.
(2) Looking at a graph or table we surmise that 1 might be the upper bound. Performing synthetic division we get 1 7 7 5 1; all nonnegative so 1 is an upper bound on the real roots.
Thus a lower bound is -6, and an upper bound is 1.
** You could repeat the process using rational numbers to narrow your search for the zero -- but in this case the real zeros are very close to -6 and 1; the zeros are `x~~-5.96261` and `x~~.946661`
See reference link for a thorough description of synthetic division.