To find the largest negative integer which is a lower bound on the set of negative zeros of \(f(x)=x^3-5x+2\) we must find the smallest positive integer which is a upper bound on the reflection of the graph of \(y=f(x)\) in the \(y\)-axis. That is, we must find the lowest integral bound on the graph of \(y=f(-x)\).
\(f ( - x ) = - x^3 + 5 x + 2\).
If we find the least integral upper bound \(c\) on the set of positive zeros of \(f ( - x )\), then \(- c\) will be the greatest integral lower bound on the set of negative zeros of \(f ( x )\). But for this particular polynomial \(- x^3 + 5 x + 2\) the leading coefficient is negative, so we will never get a row of all non-negative numbers since the second entry in each row is always the leading coefficient of the polynomial.
So we look for the bounds on the zeros of \(- f ( -x )\), since \(- f ( -x )\) has the same zeros as \(f (- x )\).
\(- f ( - x ) = x^3 - 5 x - 2\) with a positive leading coefficient.
So we use synthetic division to search for the least postive integral upper bound on \(x^3 - 5 x - 2\).
| \(1\) | \(0\) | \(-5\) | \(-2\) | |
| \(1\) | \(1\) | \(1\) | \(-4\) | \(-6\) |
| \(2\) | \(1\) | \(2\) | \(-1\) | \(-4\) |
| \(3\) | \(1\) | \(3\) | \(4\) | \(10\) |
Since \(3\) is the smallest positive integer which gives a row of all non-negative integers, \(3\) is the least integral upper bound on the zeros of \(x^3 - 5 x - 2\). It follows that \(-3\) is the greatest integral lower bound on the negative zeros of \(f(x)=x^3-5x+2\).