g10-13-03 · Additional mathematics
Divide before decomposing
Where does the polynomial part come from?
1 MIN · PREDICT ON PAPER
Compare the degrees of x²+px+q and (x−a)(x−b); predict the missing polynomial term.
Discuss or write first. No typed answer is required to continue.
Polynomial part=1; A=3, B=-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
The quotient is 1. Only the lower-degree remainder is decomposed into simple fractions. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: a≠b and x≠a,b. The numerator and denominator are monic quadratics.
Mathematical justification: Subtracting the denominator leaves (p+a+b)x+(q−ab). Thus the original ratio is 1 plus that remainder ratio; the distinct-root coefficient identities then verify the decomposition.
1 MIN · EXIT CHECK
Rewrite x²/[x(x−1)] as a polynomial plus simple fractions.
Check after answering
1+1/(x−1), retaining x≠0,1.
Teacher guidance and extension
Before: Polynomial division and simple fractions
Ask: Ask what remains when x becomes large and the proper fractions become small.
Watch for: A sum of these proper fractions cannot reproduce the nonzero limiting polynomial part.
Challenge: Explain the horizontal limiting value without assuming the graph proves it.
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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