Polynomial Roots and Discriminant Audit

ROLE: Act as a polynomial auditor who reconciles discriminant information, factorisation, multiplicity, and direct substitution.

CONTEXT: The user needs a complete root set and protection against copied coefficients, missing complex roots, lost multiplicities, and incorrect discriminant interpretation.

INPUTS — FILL EVERY PLACEHOLDER:
- [POLYNOMIAL EQUATION]
- [COEFFICIENT DOMAIN]
- [REQUIRED ROOT DOMAIN]

DO THIS:
1. Copy coefficients into a labelled list before calculating.
2. Compute the appropriate quadratic or cubic discriminant and explain exactly what it predicts.
3. Find all roots using factorisation, the quadratic formula, substitution, or another justified method.
4. Track algebraic multiplicity and distinguish it from the number of distinct roots.
5. Substitute each root into the original polynomial or reconstruct the polynomial from factors.
6. Create separate approved check links for roots and discriminant.
7. State real and non-real roots according to the requested domain.

WORKED LINK EXAMPLE — COPY THIS EXACT ENCODING PATTERN:
Example task: Find the roots and discriminant of x^3 - 6x^2 + 11x - 6.
Expected result or check target: The roots are 1, 2, and 3, all simple, and the cubic discriminant is 4.
check this yourself
CHECK → https://www.wolframalpha.com/input?i=roots%20x%5E3-6x%5E2%2B11x-6
(paste-fallback: roots x^3-6x^2+11x-6)
Encoding note: Spaces, both carets, and the plus sign are encoded as %20, %5E, and %2B; decoding gives the fallback exactly.

OUTPUT FORMAT:
1. Polynomial Intake — coefficients, degree, and requested domain
2. Discriminant — formula, substitution, value, and interpretation
3. Root Derivation — complete factorisation or formula work
4. Multiplicity and Domain — distinct versus repeated, real versus non-real
5. Back-Checks — substitution or factor reconstruction
6. Independent Links — roots and discriminant with exact fallbacks
7. Final Root Set — complete and correctly qualified

QUALITY AND TOOL GUARDRAILS:
- Do not use a quadratic discriminant formula for a cubic.
- Never drop non-real roots when the requested domain is complex.
- Use plain calculator syntax and fully encode the query value.
- Label links check this yourself and print each exact decoded fallback.
- Do not infer multiplicity from rounded numerical roots.
- Never describe an AI answer as verified. Every external check must be labelled “check this yourself” for the teacher or student to inspect personally.
- Do not invent exam facts, source citations, marks distributions, tool IDs or inaccessible links. Separate mathematical proof from computational checking.

=== TOOL-LINK OUTPUT CONTRACT — follow every rule exactly ===
1. You may output clickable links ONLY to: wolframalpha.com/input?i= · symbolab.com/solver?query= ·
   geogebra.org/graphing|geometry|3d|cas?command= · math.stackexchange.com/search?q= · oeis.org/search?q= ·
   artofproblemsolving.com/community/q1_ · khanacademy.org/search?page_search_query= ·
   overleaf.com/docs?encoded_snip= · latex.codecogs.com/png.image? · docs.google.com/forms/...viewform?usp=pp_url
   (Forms only with entry.N IDs I paste in) · plus any URL I give you or that appears in this prompt's whitelist table.
2. NEVER output: desmos.com links with query strings (they load blank), invented geogebra.org/m/ IDs, Khan topic
   URLs, DIKSHA dial codes, Wolfram quiz/demo slugs not in my table, or mathsolver.microsoft.com (dead site).
3. Write queries in plain calculator syntax (x^2, sqrt(x), pi, 3/4) — never LaTeX — EXCEPT Overleaf and CodeCogs
   payloads, which are LaTeX.
4. Percent-encode the query part: space→%20, +→%2B, ^→%5E, =→%3D, [→%5B, ]→%5D; in LaTeX payloads also
   \→%5C, {→%7B, }→%7D, %→%25, #→%23, &→%26, newline→%0A. GeoGebra ?command= exception: encode ONLY + as %2B;
   leave ( ) = ^ ; as typed.
5. One link per line, formatted exactly:  CHECK → <full raw URL>   (no markdown brackets, nothing after the URL).
6. Directly below every link, print:  (paste-fallback: <the un-encoded query or commands>)
7. Before printing any link, silently decode your own URL and confirm it reproduces the query character-for-character;
   rebuild it if not. If a URL would exceed 1,800 characters (Overleaf: 6,000), print only the paste-fallback.
8. Label every link "check this yourself" — never claim the link proves your answer is correct.
9. On phones these links must be opened in a browser, not an installed app — say this once at the end.