789 prompts matching "Structured question: polynomial with a given remainder pattern"
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Explain back: completing the square when the coefficient of x squared is not one
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Worked Example: Factorising a quadratic whose leading coefficient is not one
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Explain Back: The zero product rule, and why it works
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Show All Working: A quadratic whose two roots are both negative
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Common Mistakes: Writing down the roots once the brackets are found
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Worked Example: Halving the coefficient of x and squaring it
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Practice: Working through a quadratic by completing the square, step by step
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Explain Back: An answer left in surd form after completing the square
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Common Mistakes: Checking a completed square by expanding it again
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Worked Example: Substituting a, b and c into the quadratic formula correctly
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Practice: Giving the roots correct to two decimal places
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Explain Back: Leaving the roots of a quadratic in surd form
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Show All Working: Choosing between factorising and the formula
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Common Mistakes: A sign error in the formula, and how to catch it
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Worked Example: The product of the roots as c over a
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Explain Back: Finding a missing coefficient from a stated sum of roots
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Show All Working: Roots equal in magnitude but opposite in sign
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Common Mistakes: Roots that are reciprocals of one another
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Worked Example: The identity linking alpha squared plus beta squared to the sum and product
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Practice: The value of alpha to the fourth plus beta to the fourth
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