g10-13-02 · Additional mathematics
Why include every repeated power?
Can one term represent every numerator over (x−a)²?
1 MIN · PREDICT ON PAPER
Compare A/(x−a)+B/(x−a)² with the incomplete proposal B/(x−a)².
Discuss or write first. No typed answer is required to continue.
Repeated-factor coefficients: A=1, B=1; x≠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
For numerator mx+n, coefficient comparison gives A=m and B=ma+n. Dropping A removes an entire variable term unless m=0. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: x≠a. The repeated-root form is used even if a numerator cancellation is possible.
Mathematical justification: Clearing denominators gives mx+n=A(x−a)+B. The x coefficient forces A=m; the constant then forces B=n+ma.
1 MIN · EXIT CHECK
Decompose x/(x−1)².
Check after answering
1/(x−1)+1/(x−1)², with x≠1.
Teacher guidance and extension
Before: Coefficient comparison
Ask: Show the incorrect graph as well as the corrected graph and ask for the missing coefficient.
Watch for: One numerical match does not prove an identity of rational expressions.
Challenge: Generalise the template to denominator (x−a)³.
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.
Optional local notebook
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