Remainder Theorem

Remainder Theorem

What is it?

When p(x) is divided by x-c, the remainder is p(c). You do not need to complete the long division to know that leftover number. Meaning: plug in the root of the divisor. If you divide by ax-b, the remainder is p(ba) (still a constant). Proof idea: p(x)=(x-c)q(x)+R, put x=c, get p(c)=R.

In simple words

• When p(x) is divided by x-c, the remainder is p(c).
• You do not need to complete the long division to know that leftover number.
• Meaning: plug in the root of the divisor.
• Dividing by x+c is dividing by x-(-c). Plug in -c, not +c.

Important terms

REMAINDER THEOREM: When p(x) is divided by x-c, the remainder is p(c). You do not need to complete the long division to know that leftover number. Meaning: plug in the root of the divisor. If you divide by ax-b, the remainder is p(ba) (still a constant). Proof idea: p(x)=(x-c)q(x)+R, put x=c, get p(c)=R.

1. WHY: 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.

2. FIND REMAINDERS: p(x)=x^3-2x+5 divided by x-1: remainder p(1)=4. By x+2 i.e. x-(-2): p(-2). By 2x-1: remainder p(12).

What you should learn

• State the main ideas and notation used in remainder theorem.
• Apply the rules of remainder theorem in a clear, step-by-step solution.
• Check signs, conditions, and the final answer for mathematical accuracy.

Examples

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

WHY

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.

FIND REMAINDERS

p(x)=x³-2x+5 divided by x-1: remainder p(1)=4. By x+2 i.e. x-(-2): p(-2). By 2x-1: remainder p(12).

Full school-level explanation

When p(x) is divided by x-c, the remainder is p(c). You do not need to complete the long division to know that leftover number. Meaning: plug in the root of the divisor. If you divide by ax-b, the remainder is p(ba) (still a constant). Proof idea: p(x)=(x-c)q(x)+R, put x=c, get p(c)=R.

Why it works, then use it:

WHAT — REMAINDER THEOREM

When p(x) is divided by x-c, the remainder is p(c). You do not need to complete the long division to know that leftover number. Meaning: plug in the root of the divisor. If you divide by ax-b, the remainder is p(ba) (still a constant). Proof idea: p(x)=(x-c)q(x)+R, put x=c, get p(c)=R.

RULE — 1. WHY

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.

METHOD — 2. FIND REMAINDERS

p(x)=x³-2x+5 divided by x-1: remainder p(1)=4. By x+2 i.e. x-(-2): p(-2). By 2x-1: remainder p(12).

WORKED EXAMPLE

Worked: remainder of x^3 - 4x + 1 divided by x-2. p(2) = 8 - 8 + 1 = 1 remainder 1

WHY

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

FIND REMAINDERS

p(x)=x³-2x+5 divided by x-1: remainder p(1)=4. By x+2 i.e. x-(-2): p(-2). By 2x-1: remainder p(12).

Remember

• match x-c

Examples

• x-1 -> p(1)
• x+3 -> p(-3)
• 2x-1 -> p(12)

How to answer a remainder theorem question

How it works

• Identify the given quantities, notation, and required result.
• Choose the matching rule and write it before substitution.
• Work one algebraic step at a time, keeping brackets and signs clear.
• Check restrictions, simplify the result, and verify it where possible.

Remember

• divide by x-c: remainder p(c)
• just substitute
• 1
• x+c -> p(-c)
• p(c)=R
• match x-c
• divide by x-c: remainder p(c)
• just substitute

How the ideas connect

Worked: remainder of x^3 - 4x + 1 divided by x-2. p(2) = 8 - 8 + 1 = 1 remainder 1

Deeper understanding

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

How the ideas connect

Worked: remainder of x^3 - 4x + 1 divided by x-2. p(2) = 8 - 8 + 1 = 1 remainder 1

Common mistakes

Exam focus

Core Idea

1. WHY

2. FIND REMAINDERS

Worked Example

Exam focus

Quick revision

• divide by x-c: remainder p(c)
• just substitute
• 1
• x+c -> p(-c)
• REMAINDER THEOREM: When p(x) is divided by x-c, the remainder is p(c). You do not need to complete the long division to
• 1. WHY: Identity p(x)=(x-c)q(x)+R. At x=c the first piece dies. R is constant because the divisor is degree

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

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