Remainder theorem and factor theorem: Class 9 practice
When p(x) is divided by (x − a), the remainder is p(a). Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Evaluating expressions with negatives
The idea behind remainder theorem and factor theorem
When p(x) is divided by (x − a), the remainder is p(a). If p(a) = 0, then (x − a) is a factor of p(x).
Step-by-step method
- Set the divisor to zero to find a.
- Substitute a into p(x).
- Remainder = p(a); a factor if it is 0.
Worked example 1 (Basic)
Find the remainder when 2x³ + 4x² + 6x − 5 is divided by (x − 1).
- Put x = 1
- p(1) = 2 + 4 + 6 − 5 = 7
- Answer: 7
Worked example 2 (Standard)
Find the remainder when 1x³ − 3x² − 3x − 5 is divided by (x + 2).
- Put x = -2
- p(-2) = -8 − 12 + 6 − 5 = -19
- Answer: -19
Worked example 3 (Challenge)
Find k if (x + 3) is a factor of 3x³ + 2x² − 6x + k.
- p(-3) = 0
- -81 + 18 + 18 + k = 0
- k = 45
- Answer: k = 45
Common mistakes and misconceptions
- Substituting −a for (x − a)
- Sign errors with odd powers of negative numbers
Practice questions
- (Basic) Find the remainder when 2x³ − 2x² + 6x + 3 is divided by (x − 2).
- (Basic) Find the remainder when 3x³ + 0x² − 4x − 9 is divided by (x − 3).
- (Standard) Find the remainder when 1x³ + 4x² + 1x + 6 is divided by (x + 3).
- (Standard) Find the remainder when 2x³ − 4x² + 4x + 7 is divided by (x − 2).
- (Challenge) Find k if (x − 1) is a factor of 1x³ − 2x² − 4x + k.
- (Challenge) Find k if (x − 2) is a factor of 3x³ − 1x² + 2x + k.
Answer key with full working
- Q1: Put x = 2 → p(2) = 16 − 8 + 12 + 3 = 23 → Answer: 23
- Q2: Put x = 3 → p(3) = 81 + 0 − 12 − 9 = 60 → Answer: 60
- Q3: Put x = -3 → p(-3) = -27 + 36 − 3 + 6 = 12 → Answer: 12
- Q4: Put x = 2 → p(2) = 16 − 16 + 8 + 7 = 15 → Answer: 15
- Q5: p(1) = 0 → 1 − 2 − 4 + k = 0 → k = 5 → Answer: k = 5
- Q6: p(2) = 0 → 24 − 4 + 4 + k = 0 → k = -24 → Answer: k = -24
Syllabus note
This skill is listed under "Polynomials" for Class 9 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is remainder theorem and factor theorem in?
"Polynomials", Class 9 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.