We perform abbreviated synthetic division with each positive integer beginning with \(1\) until we find one which results in a row of non-negative numbers.
| \(1\) | \(-3\) | \(-3\) | \(1\) | |
| \(1\) | \(1\) | \(-2\) | \(-5\) | \(-4\) |
| \(2\) | \(1\) | \(-1\) | \(-5\) | \(-9\) |
| \(3\) | \(1\) | \(0\) | \(-3\) | \(-8\) |
| \(4\) | \(1\) | \(1\) | \(1\) | \(5\) |
Since \(4\) is the smallest positive integer resulting in a row of non-negative numbers, \(4\) is the smallest integral upper bound on the positive zeros of the function \(f(x)=x^3-3x^3-3x+1\).
Notice that from the synthetic division we see that \(f(3)=-8\) and \(f(4)=+5\). So the graph must cross the \(x\)-axis between \(3\) and \(4\) which means that the function has a zero between \(3\) and \(4\) but no zeros larger than \(4\) since \(4\) is an upper bound on the positive zeros of the function.