Rational root theorem.
Test rational candidates from factors of constant and leading coefficient.
Setup
AI Solver by AskSia
Find the Zeros Solver: roots of any polynomial
Type or photograph the polynomial. AskSia finds all zeros (real and complex), using rational root theorem, factoring, the quadratic formula, or numerical methods.
Works with word problems, equations, code, and science prompts.
∫ 3x² · sin(x) dx
SubjectsCalculusAlgebraPhysicsChemistryBiologyCSStatisticsEcon
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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
What are the zeros of a polynomial?
The zeros of a polynomial p(x) are the values of x for which p(x) equals zero. Equivalently, they are the x-intercepts of the graph of y equals p(x). A polynomial of degree n has exactly n zeros (counted with multiplicity) in the complex numbers; some may be real and some complex (complex zeros come in conjugate pairs for polynomials with real coefficients). Finding zeros uses the rational root theorem to test rational candidates, synthetic division to reduce, and quadratic formula or numerical methods for the final step.
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solution accuracy
50M+
problems solved
~1.5s
avg solve time
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study-ready explanations
Why AskSia Solver
Why students use AskSia for Find the Zeros.
Every step transparent, every answer self-checked.
Synthetic division.
Divide out each found root to reduce the polynomial.
Reduction
Quadratic formula.
Apply to the final quadratic factor for the last two zeros.
Closing
Complex zeros.
Complex conjugate pairs identified when present.
Complex
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 Find the Zeros 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
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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
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Calculus
98% verified
1.4s
Step 4 of 4 · Evaluate
A = [x² - x³/3]₀² = 4/3
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1.4s
Area between y=2x & y=x²
A = 4/3 sq. units ✓
Use Cases
What the Find the Zeros solver covers.
Factor a polynomial.
Find rational roots, divide out, factor remaining quadratic.
Factor
Multiplicity tracking.
A zero may appear multiple times; AskSia reports multiplicities.
Multiplicity
Complex zeros.
Quadratic with negative discriminant gives a complex conjugate pair.
Complex
Graphing aid.
Zeros are x-intercepts; multiplicities determine touching vs crossing.
Graphing
Numerical roots.
When rational roots fail, numerical methods give decimal approximations.
Numerical
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 zeros does a polynomial have?
A polynomial of degree n has exactly n zeros in the complex numbers, counted with multiplicity. For a quadratic, two zeros; cubic, three; quartic, four. Some may be repeated (multiplicities greater than 1) and some may be complex. For polynomials with real coefficients, complex zeros come in conjugate pairs, so the number of complex zeros is even.
What is multiplicity?
Multiplicity counts how many times a zero appears. For example, if p(x) equals (x minus 2) squared times (x plus 3), the zero x equals 2 has multiplicity 2 (counted twice) and x equals minus 3 has multiplicity 1. Multiplicity affects the graph: at a zero of even multiplicity, the curve touches but does not cross the x-axis; at odd multiplicity, it crosses.
How does the rational root theorem help?
It narrows the search for rational zeros to a finite list: rational zeros p over q (in lowest terms) must have p as a factor of the constant term and q as a factor of the leading coefficient. Testing this finite list by substitution or synthetic division reveals which candidates are actual zeros. Other zeros (irrational, complex) are found by other methods after rational ones are divided out.
What if no rational zeros exist?
Some polynomials have only irrational or complex zeros (e.g., x squared plus 1 has zeros plus or minus i, no rational ones). In these cases, the quadratic formula applies if you reduce to a quadratic. For higher-degree polynomials without rational zeros, numerical methods (Newton's method, bisection) approximate the roots; exact forms involve nested radicals (cubic, quartic) or do not exist in elementary functions (quintic and higher).
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.
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Rational roots, division, formula.
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