Proof of the Remainder Theorem

Proof of the Remainder Theorem

What is it?

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.

In simple words

• 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.

Important terms

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.

What you should learn

• 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.

Examples

• No long division needed
• c is the 'plug-in'
• R a number

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.

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.

Full school-level explanation

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.

Understand the idea, then apply it:

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.

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.

Remember

• p(c)=R

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.

Remember

• focused answer
• accurate support

Examples

• State the central idea
• Use one accurate detail
• Check the final response

How to answer a proof of the remainder theorem question

How it works

• 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.

Remember

• p(c)=R
• divide by x-c: remainder p(c)
• just substitute
• x+c -> p(-c)
• p(c)=R
• focused answer
• accurate support
• p(c)=R

Deeper understanding

Go beyond memorising Proof of the Remainder Theorem. Connect the definition to reasons, relationships, exceptions, and exam wording.

Common mistakes

Exam focus

Core Idea

What to Know

Example or Evidence

How to Answer

Exam focus

Quick revision

• p(c)=R
• divide by x-c: remainder p(c)
• just substitute
• x+c -> p(-c)
• 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
• WHY: Identity p(x)=(x-c)q(x)+R.

Dividing by x+c is dividing by x-(-c). Plug in -c, not +c.

For long answers on Proof of the Remainder Theorem: define clearly → explain the main parts → give one example → state one exam caution.