g10-04-02 · Core topics
Where did the missing terms go?
Can missing coefficients be omitted in synthetic division?
1 MIN · PREDICT ON PAPER
For x⁴+bx²+c, predict the five coefficient slots before dividing by x−a.
Discuss or write first. No typed answer is required to continue.
Division identity: f(x)=(x−(1))(x^3 +x^2 −2x −2)+(0)
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
Missing x³ and x terms have coefficient zero. Keeping their slots preserves the powers during multiply-and-add steps. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: The polynomial has degree four even when b or c is zero. Division is by x−a.
Mathematical justification: Horner recurrence h₀=1, hᵢ=cᵢ+a hᵢ₋₁ matches successive coefficient cancellation. With zero slots retained, expansion recovers the original coefficients.
1 MIN · EXIT CHECK
List the coefficient row for x⁴−3x²+2.
Check after answering
[1,0,−3,0,2], including both missing terms.
Teacher guidance and extension
Before: Polynomial coefficient order
Ask: Ask pupils to label the power above every coefficient before moving counters.
Watch for: Removing a zero slot changes the polynomial rather than simplifying its notation.
Challenge: Compare the erroneous division using [1,b,c] with the correct five-slot computation.
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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