g10-12-01 · Additional mathematics
Which factor copies are shared?
Why does a greatest common divisor use the lower exponent?
1 MIN · PREDICT ON PAPER
Build A=(x−1)ᵐ(x+1)ⁿ and B=(x−1)ᵖ(x+1)ᑫ; predict their shared tiles.
Discuss or write first. No typed answer is required to continue.
Monic result: GCD = (x−1)^1(x+1)^1
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
A common divisor cannot use more copies than either polynomial contains, so it uses the minimum exponent of each factor. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: Both polynomials are monic and nonzero. Exponent zero contributes the factor 1.
Mathematical justification: For each distinct irreducible factor, a common divisor exponent is at most both source exponents, hence at most their minimum. The constructed product itself divides both.
1 MIN · EXIT CHECK
Find the monic GCD of (x−1)²(x+1) and (x−1)(x+1)³.
Check after answering
(x−1)(x+1).
Teacher guidance and extension
Before: Factorisation and divisibility
Ask: Request a divisibility check by subtracting factor counts.
Watch for: Greatest refers to divisibility, not whichever evaluated number is largest.
Challenge: Show that every common divisor divides the displayed greatest common divisor.
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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