📉 Class 9 · Algebra

Polynomials for Class 9: notes, worked examples and practice

A polynomial is a sum of terms like 3x² − 5x + 2. This chapter teaches you to evaluate them, find their zeros and use the remainder and factor theorems.

Updated 21 October 2026. Every worked example and answer on this page is checked by automated tests.

Try the practice questions WhatsApp a maths teacher

Notes

Polynomials: what you need to know

Chapter names and order differ by board and textbook edition, so follow your own textbook for the order.

What a polynomial is

A polynomial in one variable is a sum of terms with whole-number powers of x, such as 2x² − 3x + 1. The highest power is the degree. A degree 1 polynomial is linear, degree 2 is quadratic and degree 3 is cubic.

Value and zeros

p(a) is the value of p(x) when x = a. If p(a) = 0, then a is a zero of the polynomial. A polynomial of degree n has at most n zeros.

Remainder theorem

When p(x) is divided by (x − a), the remainder is p(a). To find a remainder you only need to evaluate, not divide.

Factor theorem

(x − a) is a factor of p(x) exactly when p(a) = 0. Use it to test whether a given expression is a factor and to factorise cubics.

Algebraic identities

(a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b² and a² − b² = (a + b)(a − b). They speed up expansion and factorisation.

Formulas and rules to remember

Remainder when p(x) ÷ (x − a) is p(a)
(x − a) is a factor of p(x) if and only if p(a) = 0
(a + b)² = a² + 2ab + b²
a² − b² = (a + b)(a − b)

Solved examples

Example 1

If p(x) = 2x² − 3x + 1, find p(2).

p(2) = 2 × 2² − 3 × 2 + 1.
= 8 − 6 + 1 = 3.

Answer: 3

Example 2

Find the zeros of x² − 5x + 6.

Factorise: x² − 5x + 6 = (x − 2)(x − 3).
The zeros are x = 2 and x = 3.

Answer: 2 and 3

Example 3

Find the remainder when x³ − 2x² + x − 5 is divided by x − 3.

By the remainder theorem the remainder is p(3).
p(3) = 27 − 18 + 3 − 5 = 7.

Answer: 7

Example 4

Is (x − 1) a factor of x³ − 6x² + 11x − 6?

Find p(1) = 1 − 6 + 11 − 6.
p(1) = 0, so (x − 1) is a factor by the factor theorem.

Answer: Yes, because p(1) = 0

Avoid these

Common mistakes

Using the wrong sign of a

For a divisor (x + 2), the value to evaluate is p(−2), because x + 2 = x − (−2).

Mixing up degree and number of terms

x³ + 1 has degree 3 but only two terms.

Confusing zeros with coefficients

A zero is a value of x that makes the polynomial 0. Coefficients are the numbers in front of the powers.

10 practice questions with answers

Try each question on paper first, then tap it to check your answer. The questions are generated by code and each answer is checked.

If p(x) = 3x² + 8, find p(4)
Answer: 56
Find the zeros of x² − 14x + 48
Answer: 6 and 8
Find the remainder when x² + 3x + 1 is divided by x + 3
Answer: 1
If p(x) = 3x² + 4x + 3, find p(−3)
Answer: 18
Find the zeros of x² − 9x + 18
Answer: 3 and 6
Find the remainder when x² + 8 is divided by x − 2
Answer: 12
Find the zeros of x² − 3x + 2
Answer: 1 and 2
Find the zeros of x² − 5x + 6
Answer: 2 and 3
Find the zeros of x² − 10x + 21
Answer: 3 and 7
If p(x) = x² + 6x + 3, find p(4)
Answer: 43

More practice

Printable worksheets (PDF with answers)

Evaluating algebraic expressions (Grade 6)

Games and lessons

Online maths practice

Maths dictionary

Quadratic equation solver

Solve ax² + bx + c = 0 with the discriminant, both roots (including complex ones), the vertex and the factorised form, step by step.

Use the tool →

FAQ

Polynomials Class 9: questions students ask

What is the degree of a polynomial?

The highest power of the variable. 5x³ + 2x − 1 has degree 3.

What is the remainder theorem?

When p(x) is divided by (x − a) the remainder is p(a).

How do I use the factor theorem?

Evaluate p(a). If it is zero then (x − a) is a factor.

What is a zero of a polynomial?

A value of x for which the polynomial equals zero.

Which board is this polynomials page for?

It covers the ideas taught in CBSE, ICSE and state-board maths. Chapter names and order differ by board and textbook edition, so follow your own textbook for the exact order.

Need help with polynomials?

Tell us the class, board and what feels hard. We match a maths teacher from our network and confirm the format before anything is booked.

Tell us the learner’s age or grade and goals
We’ll suggest a suitable class or learning path
We’ll confirm format, timing and fees with a parent or guardian

Play it, don't just read it

Try it now: Evaluating algebraic expressions

Eight quick questions with instant feedback. Your score is saved on this device and counts in the games and learning paths too.

All free games →

Evaluating algebraic expressions (Grade 6): Substituting a value for a letter and working out the answer. Open the practice page or print a worksheet.

Or play a game

Today's maths challengeFive questions · the same for everyone today · keep your streak goingPlay