MathematicsChapter 5: Polynomials
Proof of the Remainder Theorem
Identity p(x)=(x-c)q(x)+R.
Learning objectives
- State the rule or meaning of proof of the remainder theorem in easy words.
- Apply proof of the remainder theorem in a clear step-by-step solution.
- Check the conditions, working, and final answer for accuracy.
Definition
Identity p(x)=(x-c)q(x)+R. At x=c the first piece dies. R is constant because the divisor is degree 1. That is the whole theorem.
Formula
x=c the first piece dies. R is constant be
Where / symbols
- PROOF OF THE REMAINDER THEOREM
- Identity p(x)=(x-c)q(x)+R. At x=c the first piece dies. R is constant because the divisor is degree 1. That is the whole theorem.
- WHY
- Identity p(x)=(x-c)q(x)+R.
Relationship
- p(c)=R
Overview
Understand the idea, then apply it:
Main Idea
Identity p(x)=(x-c)q(x)+R. At x=c the first piece dies. R is constant because the divisor is degree 1. That is the whole theorem.
Examples No long division needed · c is the 'plug-in' · R a number
Exam Use
Write the rule first: Identity p(x)=(x-c)q(x)+R. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Examples State the central idea · Use one accurate detail · Check the final response
WHAT — PROOF OF THE REMAINDER THEOREM
Identity p(x)=(x-c)q(x)+R. At x=c the first piece dies. R is constant because the divisor is degree 1. That is the whole theorem.
WHY — Why it matters
This rule is a separate learning unit inside remainder theorem. It can be defined, applied, and checked in its own FBISE question.
HOW — How to answer
Write the rule first: Identity p(x)=(x-c)q(x)+R. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
How to answer a proof of the remainder theorem question
- Read the exact command word and identify what the question asks.
- State the matching idea, rule, or meaning before adding detail.
- Use the unit evidence or mathematical working that directly supports the answer.
- Check accuracy, relevance, and answer length before finishing.
In simple words
Dividing by x+c is dividing by x-(-c). Plug in -c, not +c.
Key points
- p(c)=R
- divide by x-c: remainder p(c)
- just substitute
- focused answer
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Proof of the Remainder Theorem
Sticky Notes · Quick Revision
Core Idea
- Identity p(x)=(x-c)q(x)+R.
What to Know
- Identity p(x)=(x-c)q(x)+R. At x=c the first piece dies. R is constant because the divisor is degree 1. That is the whole theorem.
Example or Evidence
- No long division needed
- c is the 'plug-in'
- R a number
How to Answer
- Write the rule first: Identity p(x)=(x-c)q(x)+R. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Common Error
- Dividing by x+c is dividing by x-(-c). Plug in -c, not +c.
FBISE Exam Check
- State the rule, show the working, and check the answer.
60-second revision
- Identity p(x)=(x-c)q(x)+R.
- No long division needed
- c is the 'plug-in'
- State the rule, show the working, and check the answer.
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