789 prompts matching "Structured question: polynomial with a given remainder pattern"
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Explain Back: The value of alpha minus beta, all squared
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Show All Working: Forming an equation whose roots are three alpha and three beta
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Common Mistakes: A symmetric function needing the sum and product twice over
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Worked Example: Substituting into the quadratic and tidying up
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Practice: Finding the matching value of the second unknown
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Explain Back: Writing the answers as ordered pairs
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Show All Working: A simultaneous pair in which the second equation is a circle
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Common Mistakes: A word problem giving one linear and one quadratic equation
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Worked Example: The discriminant used as the test for intersection
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Practice: The condition for the line to be a tangent
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Explain Back: The condition for the line never to meet the curve
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Show All Working: The point of contact when a line touches a curve
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Show All Working: Synthetic division as a shorter route
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Worked Example: Testing a list of candidates to find a factor
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Explain Back: Finding a coefficient so that a stated expression is a factor
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Show All Working: What the factor theorem says about the roots
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Common Mistakes: Using the factor theorem before dividing
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Worked Example: Factorising a cubic into a linear and a quadratic factor
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Practice: Factorising a quartic as the product of two quadratics
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Common Mistakes: Factorising fully, and knowing when to stop
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