g10-04-03 · Core topics
Read a remainder from one point
Why is the remainder f(a)?
1 MIN · PREDICT ON PAPER
Divide f(x)=x³+bx+c by x−a; predict which graph input reveals the remainder.
Discuss or write first. No typed answer is required to continue.
Division identity: f(x)=(x−(2))(x^2 +2x +2)+(7)
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
At x=a the quotient term is multiplied by zero, leaving f(a)=r. The sign of a comes from the complete divisor. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: The divisor must be x−a; division by x+a uses input −a.
Mathematical justification: The division identity f=(x−a)q+r is a polynomial identity, so substitution x=a is valid and gives r=f(a) exactly.
1 MIN · EXIT CHECK
Find the remainder of x³+1 divided by x+2.
Check after answering
f(−2)=−8+1=−7.
Teacher guidance and extension
Before: Division identity and substitution
Ask: Cover the long-division steps and ask why the marked ordinate is sufficient.
Watch for: A point with small output is not necessarily an exact zero.
Challenge: Recover the remainders for x−2 and x+2 without performing long division.
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
Paper responses are enough. Drafts remain in this tab; choose local saving to include them in the existing complete backup.
Complete backup and restore