AtlasCalcsUtility Calculation Suite
2026-09-29 9 min

Polynomial Inequality Calculator

Free polynomial inequality calculator: solve ax³ + bx² + cx + d <, <=, >, >= 0 with numeric real roots, sign chart and intervals.

Calculator

Cubic Coefficients

Coefficient a (x³)
Coefficient b (x²)
Coefficient c (x)
Constant d
Inequality operator
Results update live as you type
Status:1x³ + -6x² + 11x + -6 < 0(-inf, 1) U (2, 3)

Solution Summary

Inequality1x³ + -6x² + 11x + -6 < 0
Real roots1, 2, 3
Solution(-inf, 1) U (2, 3)

Results

Optimal

Solution set

(-inf, 1) U (2, 3)

Union of matching sign regions

Real roots found

1, 2, 3

Bisection on sign-change brackets

Distinct real roots

3

A cubic has 1 or 3 counting multiplicity

End behavior

-inf to +inf

Sign of the leading coefficient rules the tails

Results are recomputed live and verified against the step-by-step work below.

Visualizations

Roots located

3 distinct real root(s)100%

Formula

A polynomial changes sign only at real roots of odd multiplicity, so locating the real roots settles every region.

ax³ + bx² + cx + d ? 0 — roots split the line into sign regions

Alternative Form

Odd multiplicity roots flip the sign; even multiplicity roots touch without flipping

Variables

SymbolNameUnitDescription
aCubic coeff anumberNonzero leading coefficient
b, c, dLower coeffsnumbersQuadratic, linear and constant terms
?Operatorchoice<, <=, > or >=
rootsReal rootsnumbersFound numerically by bisection

Explanation

Locate all distinct real roots by scanning for sign changes and bisecting, then test each region and keep the ones matching the operator.

Derivation

Between consecutive distinct real roots the polynomial cannot cross zero, so one test point per interval determines the whole region.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Write the cubic inequality.

ax³ + bx² + cx + d ? 0
stmt:1x³ + -6x² + 11x + -6 < 0
Result: Statement recorded
Step 2

Scan for sign changes and bisect each bracket to locate real roots.

Bisection on brackets
roots:1, 2, 3
Result: 3 distinct root(s)
Step 3

Evaluate one test point inside every region between roots.

f(sample)
n:4
Result: Sign per region known
Step 4

Keep regions whose sign matches < 0.

Solution
iv:(-inf, 1) U (2, 3)
Result: (-inf, 1) U (2, 3)
Step 5

Confirm each kept region with a second test point.

Recheck
iv:(-inf, 1) U (2, 3)
Result: (-inf, 1) U (2, 3)
Final Answer
(-inf, 1) U (2, 3)

Worked Examples

See how the calculator works with different scenarios.

Scenario

Roots at 1, 2, 3.

Inputs

a:1
b:-6
c:11
d:-6
op:<

Calculation

Negative on (-inf,1) and (2,3)

Result

(-inf, 1) U (2, 3)

Interpretation

Sign alternates across the three simple roots.

Understanding Your Result

What This Means

The headline result is the exact answer for the polynomial inequality calculator inputs you entered, with supporting values shown beside it.

Why It Matters

Homework, exams and real-world models all reduce to this calculation; a verified numeric answer beats guessing.

Professional Interpretation

Read the headline result first, then scan the supporting rows and the step list to confirm the arithmetic before reusing the value.

Recommendations

  • •Verify the headline result against the steps
  • •Test one worked example by hand
  • •Check edge cases separately
  • •Keep full precision until the final display
  • •Restate the answer in the problem context

Educational Content

Introduction

This polynomial inequality calculator evaluates your inputs with closed-form arithmetic and shows every intermediate value.

Concept Explanation

Locate all distinct real roots by scanning for sign changes and bisecting, then test each region and keep the ones matching the operator.

Applications

▸Homework and exams
▸Engineering checks
▸Finance models
▸Science labs
▸Programming logic
▸Everyday estimation

Benefits

  • •Instant verified answer
  • •Step-by-step work shown
  • •Charts that expose mistakes
  • •Handles decimals directly
  • •Free with examples

Limitations

  • •Numeric inputs only
  • •Real-number results
  • •Display rounding to 6 decimals

Best Practices

  • •Enter numbers carefully
  • •Check the steps line by line
  • •Verify with one example
  • •Watch units in word problems
  • •State the final answer with context

Common Misconceptions

Myth: One input is enough.
Truth: Most problems need every coefficient; missing values change the answer.
Myth: Rounding early is fine.
Truth: Keep full precision until the final display to avoid drift.
Myth: The chart is decoration.
Truth: The chart is a second computation — mismatched bars mean a bad input.
Myth: Any rearrangement works.
Truth: Only equivalent operations preserve equality; follow the steps.

Practical Tips

  • •Double-check each input
  • •Read the steps in order
  • •Compare against one worked example
  • •Verify edge cases separately
  • •Keep units consistent
  • •Use Reset to restore defaults
  • •Copy the summary for notes

Common Mistakes

Typing a value in the wrong field
Match each label to the formula variables before calculating.
Dividing by zero
Check denominators and leading coefficients first.
Ignoring the steps
The step list shows exactly where a sign or arithmetic slip hides.
Rounding mid-calculation
Round only the displayed answer.
Forgetting to verify
Substitute the answer back into the original statement.

Assumptions

This calculator makes the following assumptions to provide estimates:

Real numbers: Inputs and outputs are real numbers.
Exact inputs: Coefficients are taken as given.
Standard notation: Conventional symbols and operator precedence.
Display rounding: Six decimals in outputs.

Limitations

Be aware of these limitations when using this calculator:

Numeric scope: Symbolic parameters need the advanced pages.
Single scenario: One input set per calculation.
No units conversion: Convert units before entering values.

Frequently Asked Questions

Answers to common questions about this calculator.

How do you find cubic roots here?
By scanning for sign changes across a wide grid and bisecting each bracket 80 times — robust for distinct real roots without any formula memorization.
What if a root repeats?
A double root touches but may not flip the sign; test points on each side reveal the true pattern, and non-strict operators still include the root.
Why test every region?
Between consecutive distinct roots the sign cannot change, so one sample settles the whole interval.
What does multiplicity mean?
Odd multiplicity (1, 3) crosses the axis and flips sign; even multiplicity (2) touches and returns.
Can this handle degree 4+?
The same scan-and-bisect idea extends, but this page is tuned for cubics with an exact Horner evaluation.
How do I verify?
Substitute one point from each kept region back into the cubic.

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Related Guides

Deepen your understanding with these educational resources.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Solution set
All values satisfying the given statement.
Interval notation
Compact bracket notation for ranges of the number line.
Critical value
A point where the expression is zero or undefined, splitting sign regions.
Verification
Substituting the answer back to confirm the original statement.
Closed form
An exact formula evaluated directly, not an approximation loop.
Edge case
Inputs where the usual rule breaks, such as zero denominators.

Authoritative References

Trusted sources and standards referenced by this calculator.

Applicable Standards

Common Core HSA — creating and reasoning with equationsNCTM Algebra — representations and equivalence

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Page Information

Last Updated:

2026-09-29

Version:

2.0.0

License:

Free to use for educational and professional purposes.