g10-04-01 · Core topics
What does division preserve?
How can the quotient and remainder reconstruct f?
1 MIN · PREDICT ON PAPER
For a cubic divided by x−a, predict the degree of the quotient and the shape of the remainder.
Discuss or write first. No typed answer is required to continue.
Division identity: f(x)=(x−(2))(x^2)+(3)
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
Polynomial division gives f=(x−a)q+r, with constant r. The two plotted curves f and (x−a)q differ vertically by r everywhere. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: The divisor is the nonzero monic polynomial x−a; coefficients are real.
Mathematical justification: Expanding the computed quotient gives coefficient equality for every power. The remainder degree is below one, proving the identity for all real x.
1 MIN · EXIT CHECK
Divide x²+1 by x−1.
Check after answering
Quotient x+1, remainder 2; x²+1=(x−1)(x+1)+2.
Teacher guidance and extension
Before: Polynomial expansion
Ask: Point to the constant vertical difference at several positions, then ask for the coefficient proof.
Watch for: The identity holds at a too; only writing f/(x−a)=q+r/(x−a) excludes a.
Challenge: Explain why a nonconstant remainder would violate division by a linear polynomial.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
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