Rational root theorem.
Test candidate rational roots from factors of constant and leading coefficient.
First try
AI Solver by AskSia
Cubic Polynomial Solver: roots, factoring, graphing
Type or photograph the cubic. AskSia finds all roots (real and complex), factors when possible, identifies critical points, inflection points, and end behavior, and describes the graph.
Works with word problems, equations, code, and science prompts.
∫ 3x² · sin(x) dx
SubjectsCalculusAlgebraPhysicsChemistryBiologyCSStatisticsEcon
4.9 / 5 · trusted by 2M+ students · 50M+ problems solved
asksia.ai/solver
98% · Verified
Problem
Find the area enclosed between y = x² and y = 2x in the first quadrant.
1
Find the points of intersection
Set the curves equal: x² = 2x → x(x - 2) = 0, so x = 0 and x = 2.
2
Identify which curve is on top
On (0, 2), 2x > x², so y = 2x is the upper curve.
3
Set up the area integral
A = ∫₀² (2x - x²) dx
4
Integrate term-by-term
A = [ x² - x³/3 ]₀²
Final Answer
A = 4/3 sq. units
Quick Answer
How do you solve a cubic polynomial?
A cubic polynomial has the form a times x cubed plus b times x squared plus c times x plus d. To find roots, first try rational roots using the rational root theorem (factors of d over factors of a). Use synthetic division to divide out a known root, reducing to a quadratic that can be solved with the quadratic formula. If no rational roots exist, Cardano's formula gives the roots exactly, or numerical methods (Newton's, bisection) give approximations. A cubic always has at least one real root; up to three real roots are possible.
98%
solution accuracy
50M+
problems solved
~1.5s
avg solve time
A+
study-ready explanations
Why AskSia Solver
Why students use AskSia for Cubic Polynomial.
Every step transparent, every answer self-checked.
Synthetic division.
Once a root is found, divide out the factor to get a quadratic.
Reduce
Cardano's formula.
Exact roots even when irrational. AskSia uses when rational roots fail.
Exact
Graph features.
Critical points, inflection, end behavior, all derived from the cubic.
Graphing
Photo, paste, or type.
Snap handwritten or printed problems with your phone, paste from any online homework portal, or type with full LaTeX support.
Multi-modal input
Verified by AskSia.
Every answer gets a self-check pass. Sia catches sign errors and algebra mistakes before you submit your homework.
Self-checked
How It Works
Solve any Cubic Polynomial problem in three steps.
Step 01
Enter the problem.
Type the expression, paste from your homework, snap a photo, or speak it. AskSia parses your input and identifies the structure.
Input mode
Snap a Photo
Textbook, handwriting, screenshot
Paste Text
Word problem or equation
Calculator
LaTeX-ready equation editor
Step 02
AskSia picks the method.
Based on the problem structure, AskSia chooses the cleanest solution path and labels each step with the operation performed.
1
Set curves equal
x² = 2x → x = 0, x = 2
2
Set up the integral
A = ∫₀² (2x - x²) dx
3
Evaluate
A = [x² - x³/3]₀² = 4/3
Step 03
Read the verified answer.
Final result appears with a substitution or composition check. Practice problems on the same concept are one tap away.
Auto-generated diagram
Region between y = 2x and y = x² — area = 4/3
Available On
Solve anywhere
you study.
Every solve syncs across Web, iOS, and Android — start it at your desk, finish on your phone.
Web App
Full study studio
Split-panel interface with the worked solution on the left, the auto-generated diagram and AI tutor chat on the right.
Drag & drop image upload + LaTeX equation editor
Auto-generated diagrams render alongside steps
Side-panel AI tutor chat for hints and alt methods
Export to PDF, DOCX, Notion, or Google Docs
Hi! What are we studying today?
Ask about your homework, lecture, or readings...↑
Calculus
98% verified
1.4s
Step 4 of 4 · Evaluate
A = [x² - x³/3]₀² = 4/3
Mobile App
Snap & solve, anywhere
Open the camera, frame the problem, and the worked solution plus diagram appear in seconds.
One-tap snap-and-solve on iOS and Android
Pinch-to-zoom diagrams, swipe between steps
Auto-sync solves with your Web library
Offline review of saved solutions and flashcards
☰
AskSia
+What can I do for you?
Homework solver
Live transcribe
File summary
Snap
YouTube
Flashcard
Calc
98%
1.4s
Area between y=2x & y=x²
A = 4/3 sq. units ✓
Use Cases
What the Cubic Polynomial solver covers.
Factor by rational roots.
Find a root by trial, divide out, factor the resulting quadratic.
Factoring
Cardano's formula.
Exact root expressions when rational roots do not exist.
Cardano
Newton's method numerically.
Approximate roots quickly when exact form is messy.
Numerical
Critical and inflection points.
Derivative and second derivative give these.
Calculus
Sketching the cubic.
End behavior, x-intercepts, y-intercept, critical points combine for a sketch.
Graph
Verify your homework.
Paste your candidate answer and the original problem. AskSia walks the work, flags any divergent step, and tells you the correct final value.
Answer check
Compare
AskSia vs. ChatGPT,
Photomath & Symbolab.
General chatbots hallucinate. Photo solvers stop at math. AskSia is built for actual coursework with verified accuracy, visual learning, and every subject.
| Feature | AskSia Solver | ChatGPT | Photo Solvers |
|---|---|---|---|
| Solution accuracy | ✓ 98% | ~70-85%, hallucinations | ~90%, math only |
| Auto-generated diagrams | ✓ Every solve | Inconsistent / broken | Graphs only, math-only |
| Step-by-step explanations | ✓ Numbered + plain English | Inconsistent depth | ✓ Math steps |
| Subject coverage | ✓ Math, Physics, Chem, Bio, CS, Econ | ✓ Wide but unverified | Math only |
| Photo input | ✓ Handwriting + diagrams + code | Photos OK, weak on handwriting | ✓ Math photos only |
| Answer verification | ✓ Self-checked before display | No verification | Math engine only |
| Tutor follow-ups | ✓ Hints, alt methods, ELI5 | ✓ General chat | Not available |
| Practice and flashcards | ✓ One-tap from any solve | Manual prompting | Not available |
| Code debugging | ✓ Python, Java, C++, SQL... | ✓ Yes | Not available |
| Free to start | ✓ Daily solves, no card | Limited model access | Steps locked behind paywall |
FAQ
Frequently asked questions.
How many roots does a cubic have?
A cubic has exactly three roots, counted with multiplicity, in the complex numbers. At least one is real (cubics always have at least one real root because the function ranges from negative infinity to positive infinity). Two of the three may be a complex conjugate pair; or all three may be real. AskSia returns all three, real or complex, with multiplicity.
What is the rational root theorem?
If a polynomial with integer coefficients has a rational root p over q (in lowest terms), then p divides the constant term and q divides the leading coefficient. So candidate rational roots come from the factors of these two numbers. Testing each candidate by substitution or synthetic division reveals which are actual roots. The theorem reduces an infinite search to a finite list of candidates.
When should I use Cardano's formula?
When the cubic has no rational roots, and you want the exact (closed-form) roots rather than numerical approximations. Cardano's formula always works but produces complicated expressions, especially when all three roots are real (the casus irreducibilis, where the formula involves cube roots of complex numbers that combine to give real roots). For homework, AskSia uses Cardano's when needed but presents the answer cleanly.
How do you sketch a cubic from its equation?
End behavior from the leading coefficient (positive: up-up direction for large positive x and down for large negative; negative: opposite). x-intercepts from the roots; y-intercept from the constant term. Critical points (local max and min) from where the derivative is zero. Inflection point from where the second derivative is zero. Combine these landmarks for a sketch. AskSia describes each feature explicitly.
How accurate is AskSia?
AskSia is engineered for accuracy on standard high school and college coursework. Accuracy comes from subject-specialized models, a symbolic verification pass that catches arithmetic errors, and a self-check step that re-derives the answer before showing it to you.
Can I get practice problems and flashcards?
Yes. After any solve, ask Sia to generate similar practice problems at SAT, ACT, AP, IB, or college difficulty, or build a flashcard set on the underlying concept in one tap. Useful for exam prep and spaced repetition before a quiz, midterm, or final.
How much does AskSia cost?
AskSia has a free plan that includes daily solves across all subjects. AskSia Pro and Super include unlimited solves, advanced subjects, the full AI tutor companion, exports, and priority response speed. See pricing for details.
Start Today
Roots, factoring, graphing.
Join 2M+ students using AskSia to solve cubic polynomial problems step-by-step. Photo input, plain-English explanations, and a verification check on every solve.
Let's Get in Touch
