{
  "id": "LO6",
  "folder": "06_LO6_Conjugate_Root_Theorem",
  "zip": "LO6_Conjugate_Root_Theorem_SLS_xAPI.zip",
  "title": "Conjugate-Root Polynomial Detective",
  "objective": "Conjugate roots of a polynomial equation with real coefficients",
  "kind": "conjugate_roots",
  "accent": "#5f6f00",
  "stages": [
    {
      "id": "LO6-S1",
      "level": "Complete the pair",
      "prompt": "A real-coefficient polynomial has root 2+i. Which root must also occur?",
      "instruction": "Use the conjugate-root theorem.",
      "response": {
        "type": "choice",
        "options": [
          "2−i",
          "−2+i",
          "−2−i",
          "1+2i"
        ],
        "answer": "2−i"
      },
      "model": {
        "type": "roots",
        "roots": [
          {
            "re": 2,
            "im": 1
          }
        ],
        "missingMirror": true
      },
      "hint": [
        "Reflect the root across the real axis.",
        "Keep the real coordinate and reverse the imaginary coordinate."
      ],
      "worked": [
        "Real coefficients make non-real roots occur in conjugate pairs.",
        "The mirror of (2,1) is (2,−1).",
        "Therefore 2−i is also a root."
      ],
      "misconception": "Using negation rather than conjugation."
    },
    {
      "id": "LO6-S2",
      "level": "Find a coefficient",
      "prompt": "Given 2+i is a root of z³−2z²+kz+10=0 with real k, find k.",
      "instruction": "Use the required conjugate and the third root.",
      "response": {
        "type": "number",
        "answer": -3,
        "label": "k"
      },
      "model": {
        "type": "roots",
        "roots": [
          {
            "re": 2,
            "im": 1
          },
          {
            "re": 2,
            "im": -1
          },
          {
            "re": -2,
            "im": 0
          }
        ]
      },
      "hint": [
        "The pair gives factor z²−4z+5.",
        "The constant term identifies the third root as −2."
      ],
      "worked": [
        "The conjugate pair gives (z−2−i)(z−2+i)=z²−4z+5.",
        "The third factor is z+2.",
        "Expanding gives z³−2z²−3z+10, so k=−3."
      ],
      "misconception": "Forgetting the conjugate factor or using z−2 as the third factor."
    },
    {
      "id": "LO6-S3",
      "level": "Factor fully",
      "prompt": "Given 2 is a root of z³+2z²−3z−10=0, find the other roots.",
      "instruction": "Choose the exact pair after division by z−2.",
      "response": {
        "type": "choice",
        "options": [
          "−2±i",
          "2±i",
          "−1±2i",
          "−2±3i"
        ],
        "answer": "−2±i"
      },
      "model": {
        "type": "components",
        "expression": "(z−2)(z²+4z+5)",
        "real": "complete the square",
        "imag": "discriminant −4"
      },
      "hint": [
        "Divide by z−2.",
        "Solve z²+4z+5=0."
      ],
      "worked": [
        "Polynomial division gives z²+4z+5.",
        "Its discriminant is 16−20=−4.",
        "The remaining roots are −2±i."
      ],
      "misconception": "Dividing by z+2 when the known root is +2."
    },
    {
      "id": "LO6-S4",
      "level": "Quartic structure",
      "prompt": "One root of 2z⁴+5z²−3z+5=0 is 1/2−(√3/2)i. Which set gives the other three roots?",
      "instruction": "Pair the conjugate, then factor the remaining quadratic.",
      "response": {
        "type": "choice",
        "options": [
          "1/2+(√3/2)i, −1/2±(3/2)i",
          "−1/2+(√3/2)i, 1/2±(3/2)i",
          "1/2+(√3/2)i, −1±i",
          "−1/2−(√3/2)i, ±3i/2"
        ],
        "answer": "1/2+(√3/2)i, −1/2±(3/2)i"
      },
      "model": {
        "type": "roots",
        "roots": [
          {
            "re": 0.5,
            "im": -0.866
          },
          {
            "re": 0.5,
            "im": 0.866
          },
          {
            "re": -0.5,
            "im": 1.5
          },
          {
            "re": -0.5,
            "im": -1.5
          }
        ]
      },
      "hint": [
        "The known pair gives z²−z+1.",
        "The quotient is 2z²+2z+5."
      ],
      "worked": [
        "Include the conjugate 1/2+(√3/2)i.",
        "Divide by z²−z+1 to obtain 2z²+2z+5.",
        "That quadratic has roots −1/2±(3/2)i."
      ],
      "misconception": "Pairing roots by negation or assuming all four roots lie on one circle."
    },
    {
      "id": "LO6-S5",
      "level": "Unknown coefficients",
      "prompt": "Given 1−2i is a root of z⁴+z³+mz²+17z+n=0, where m,n are real, which conclusion is correct?",
      "instruction": "Use the conjugate pair and compare coefficients.",
      "response": {
        "type": "choice",
        "options": [
          "m=−2, n=−5; other roots 1+2i and (−3±√13)/2",
          "m=2, n=5; other roots −1±2i",
          "m=−2, n=5; other roots ±√13",
          "m=3, n=−1; other roots 1+2i and ±i"
        ],
        "answer": "m=−2, n=−5; other roots 1+2i and (−3±√13)/2"
      },
      "model": {
        "type": "components",
        "expression": "(z²−2z+5)(z²+az+b)",
        "real": "compare z³ and z",
        "imag": "coefficients remain real"
      },
      "hint": [
        "The known pair gives z²−2z+5.",
        "Matching z³ gives a=3; matching z gives b=−1."
      ],
      "worked": [
        "Expand (z²−2z+5)(z²+az+b).",
        "The z³ coefficient gives a−2=1, so a=3; the z coefficient gives −2b+5a=17, so b=−1.",
        "Hence m=−2, n=−5, and the remaining roots solve z²+3z−1=0."
      ],
      "misconception": "Assuming the remaining roots must also be non-real or failing to compare every coefficient."
    },
    {
      "id": "LO6-S6",
      "level": "Test the theorem",
      "prompt": "A polynomial with real coefficients lists 3+2i as a simple root but not 3−2i. What follows?",
      "instruction": "Select the logically necessary conclusion.",
      "response": {
        "type": "choice",
        "options": [
          "The root list or the claim of real coefficients is incomplete",
          "The polynomial must be quadratic",
          "3+2i is actually real",
          "No conclusion is possible"
        ],
        "answer": "The root list or the claim of real coefficients is incomplete"
      },
      "model": {
        "type": "roots",
        "roots": [
          {
            "re": 3,
            "im": 2
          }
        ],
        "missingMirror": true
      },
      "hint": [
        "Apply the theorem as a consistency check.",
        "A non-real root cannot appear alone when all coefficients are real."
      ],
      "worked": [
        "Conjugating the equation leaves real coefficients unchanged.",
        "Therefore the conjugate of every non-real root is also a root with the same multiplicity.",
        "Either 3−2i was omitted or the coefficients are not all real."
      ],
      "misconception": "Treating the theorem as optional pattern recognition rather than a consequence of real coefficients."
    }
  ]
}