How this calculation works
A zero constant can introduce a root at zero. Sign changes bound only strictly positive and strictly negative roots.
Count sign changes in f(x) and f(−x), skipping zerosA worked example
x³−6x²+11x−6 has three sign changes and permits three or one positive roots.
Good to know
Possible counts decrease from each bound by even numbers. Multiplicity counts.
Reference: OpenStax Algebra and Trigonometry (opens in a new tab) ↗
Read the result
Bound the counts of positive and negative roots of a cubic with real coefficients.
The primary result is labeled “Positive root upper bound”. Keep the input definitions in view when comparing this result with another method.
Try a related calculation
Possible counts decrease from each bound by even numbers. Multiplicity counts.
For a related exercise, try the quadratic equation solver.
A few questions, answered
How can I check the default inputs by hand?
x³−6x²+11x−6 has three sign changes and permits three or one positive roots.
Which rule does this calculator apply?
Count sign changes in f(x) and f(−x), skipping zeros