Maths Extension 1 Year 11 - Module 2 - Lesson 3
The Remainder Theorem
Name
Date
Class
1. Lesson focus
Remainder theorem. When P(x) is divided by x−a, the remainder is P(a); use the sign in the divisor carefully.
Success criteria
- Apply the lesson method accurately in a routine example.
- Show each algebraic or domain step and check the result.
- Explain why the method and restrictions fit the question.
2. Worked practice
2. Find the remainder when 2x³+x−5 is divided by x+1.
3. Find the quotient and remainder when x³−2x²+x+4 is divided by x−2.
3. Challenge
If division by x−4 leaves remainder 7, what is P(4)?
4. Multiple choice
The remainder when P(x) is divided by x+3 is which value?
A. −P(3)
B. P(3)
C. P(−3)
D. P(−1/3)
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. Find the remainder when x³−2x²+x+4 is divided by x−2.
P(2)=8−8+2+4=6, so the remainder is 6.
2. Find the remainder when 2x³+x−5 is divided by x+1.
Use a=−1: P(−1)=−2−1−5=−8.
3. Find the quotient and remainder when x³−2x²+x+4 is divided by x−2.
Synthetic division gives quotient x²+1 and remainder 6: P=(x−2)(x²+1)+6.
Challenge. If division by x−4 leaves remainder 7, what is P(4)?
By the remainder theorem, P(4)=7.
Multiple choice. The remainder when P(x) is divided by x+3 is which value?
Correct answer: C. Write x+3 as x−(−3); the theorem evaluates P at −3.
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