Polynomials is a foundation chapter in Class 9: the ideas here — degree, zeroes, the remainder and factor theorems, and the algebraic identities — come back again and again in Class 10 and beyond. Once your child can find a zero, apply the factor theorem, and use identities like (a + b)(a − b) = a² − b² to shortcut a calculation, a large family of questions becomes quick and reliable. Short, regular practice is what makes these tools feel automatic.

This free printable worksheet has 12 questions from Bmathpro’s own question bank of 10,500+ validated maths questions — the same bank our enrolled students practise from. Every answer comes with a one-line explanation, so you can check your child’s method, not just the final answer.

Set aside 15–20 minutes, keep the answers hidden until the end, and let your child attempt every question — a wrong attempt tells you more than a skipped one.

What this worksheet covers

• Degree of a polynomial and evaluating p(x) at a value
• Zeroes of a polynomial
• The remainder theorem and the factor theorem
• Factorising quadratic polynomials (including splitting the middle term)
• Algebraic identities such as (a + b)(a − b) = a² − b²

Maps to the CBSE Class 9 Polynomials chapter (NCERT) and equivalent ICSE/State-board units.

Class 9 Polynomials Worksheet — 12 Questions

Warm-up

1. Find the degree of the polynomial 5x³ − 4x² + 7x − 2. (Write your answer.)

2. If p(x) = 2x² − 3x + 1, find the value of p(2). (Write your answer.)

3. Find the zero of the linear polynomial p(x) = 2x + 6. (Write your answer.)

4. What is the degree of a non-zero constant polynomial, such as p(x) = 9?
A. 1  B. undefined  C. 0  D. 9

Building up

5. If p(x) = x³ − 4x + 5, find the value of p(−1). (Write your answer.)

6. Using the remainder theorem, find the remainder when p(x) = x³ + 3x² − x + 2 is divided by (x − 1). (Write your answer.)

7. Factorise the quadratic polynomial x² + 7x + 12. (Write the factored form.)

8. Which of the following is a factor of p(x) = x² − 5x + 6?
A. (x + 3)  B. (x − 1)  C. (x − 2)  D. (x + 2)

Challenge

9. Find the value of k for which (x − 2) is a factor of p(x) = x² − 3x + k. (Write your answer.)

10. Use the identity (a + b)(a − b) = a² − b² to evaluate 103 × 97. (Write your answer.)

11. If a + b = 7 and ab = 12, find the value of a² + b². (Write your answer.)

12. Factorise the quadratic polynomial 2x² + 7x + 3. (Write the factored form.)

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Answer key with explanations

Show answers and explanations (parents — open this after your child finishes)

1. 3 — the highest power of x is 3.
2. 3 — p(2) = 8 − 6 + 1 = 3.
3. −3 — 2x + 6 = 0 → x = −3.
4. C. 0 — a non-zero constant is 9x⁰, degree 0.
5. 8 — p(−1) = −1 + 4 + 5 = 8.
6. 5 — remainder = p(1) = 1 + 3 − 1 + 2 = 5.
7. (x + 3)(x + 4) — 3 × 4 = 12, 3 + 4 = 7.
8. C. (x − 2) — x² − 5x + 6 = (x − 2)(x − 3).
9. 2 — p(2) = 0: 4 − 6 + k = 0 → k = 2.
10. 9991 — (100 + 3)(100 − 3) = 10000 − 9 = 9991.
11. 25 — (a + b)² − 2ab = 49 − 24 = 25.
12. (2x + 1)(x + 3) — split the middle term: 2x² + x + 6x + 3.

How to use this worksheet at home

• Keep it short and calm. 15–20 minutes beats an hour of pressure.
• Name the tool first. Ask your child which theorem or identity a question is testing before they start — half the marks come from picking the right method.
• Praise the method, not the marks. (We deliberately don’t put marks on our worksheets.)

FAQs

Is this worksheet free to print?
Yes — free to use and print at home, no sign-up needed.

Which syllabus does it follow?
Standard Class 9 Polynomials chapter — CBSE (NCERT), ICSE and State boards.

What is the factor theorem, simply put?
If p(a) = 0, then (x − a) is a factor of the polynomial p(x). It lets your child check a factor, or find an unknown, without long division. Question 9 uses exactly this. A free demo class shows it in action.

Where can my child get more practice like this?
Enrolled students get Math Gym, our paid practice app with 10,500+ questions — see our courses page.

Also practise: Polynomials · Number Systems · Linear Equations in Two Variables · all Class 9 worksheets.

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