g10-04-04 · Core topics
Design a factor by changing one coefficient
What constant makes x−a a factor?
1 MIN · PREDICT ON PAPER
For x³+kx+c, predict the required c before comparing the original and repaired graphs.
Discuss or write first. No typed answer is required to continue.
Required c=-3; currently a factor: false.
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
x−a is a factor iff f(a)=0, so c=−a³−ka. Adjusting c translates the graph vertically. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: The leading coefficient stays one; only the constant is repaired.
Mathematical justification: Remainder theorem gives r=a³+ka+c. The divisor is a factor exactly when r=0, which has the unique displayed solution for c.
1 MIN · EXIT CHECK
Find c so x+1 divides x³+2x+c.
Check after answering
Set f(−1)=−1−2+c=0, giving c=3.
Teacher guidance and extension
Before: Remainder theorem
Ask: Ask pupils to justify both directions of “if and only if”.
Watch for: A factor condition is an exact equality, not merely an axis crossing nearby.
Challenge: If two different roots are required, determine whether one adjustable constant is sufficient.
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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