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Rajasthan (RBSE)Class 9 Mathematics★ 5–7 marks · RBSE Class 9 Mathematics (new NCF — Ganita Prakash 9)

Introduction to Linear Polynomials

The new NCF Class 9 Maths chapter on polynomials. Terms, variables, coefficients and degree; types of polynomials (linear, quadratic, cubic); the value and zeroes of a polynomial; the Remainder and Factor Theorems; and factorising polynomials — with solved board-pattern problems.

⏱ 30 min readMedium difficulty✓ Free · NCERT-aligned

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Mind Map

The chapter at a glance

Every concept and exam-favourite branch — revisable in 5 minutes.

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Study Guide

Night-before notes

Key points, glossary and the questions most likely to show up.

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By the end of this chapter you'll be able to…

1Identify terms, coefficients, the constant term and the degree of a polynomial
2Classify polynomials by degree (linear, quadratic, cubic) and by number of terms
3Compute the value p(a) and find the zero(es) of a polynomial
4Apply the Remainder Theorem to find a remainder without division
5Use the Factor Theorem to test and find factors

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Why this chapter matters

Polynomials are the language of algebra. The Remainder and Factor Theorems you learn here are used directly in Class 10 Polynomials and Quadratic Equations, and the factorising skill underpins almost every later algebra topic. Expect a guaranteed factorisation or theorem question in the RBSE paper.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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The World of Numbers (Class 9)

Real-number coefficients and the laws of exponents.

Introduction to Linear Polynomials (RBSE Class 9 · Mathematics)

An algebraic expression is just a recipe in symbols. When the powers of the variable are whole numbers, the recipe has a special name — a polynomial — and a rich set of rules for finding its value, its roots and its factors.

RBSE note (2026-27). Class 9 uses the new NCF (Ganita Prakash 9) Mathematics textbook. Introduction to Linear Polynomials opens the algebra strand. BSER (Ajmer) sets the exam.

1. What is a polynomial?

A polynomial in one variable x is an expression of the form:

$p (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}$

where the coefficients $a_{0}, a_{1}, \dots, a_{n}$ are real numbers and the powers of x are whole numbers (0, 1, 2, …).

Term: each part separated by + or − (e.g. 3x², −5x, 7).
Coefficient: the number multiplying a power of x (in 3x², the coefficient is 3).
Constant term: the term with no variable (7 above).

$x$ and $1/ x$ are not polynomial terms — their powers (½, −1) are not whole numbers.

2. Degree and types

The degree is the highest power of the variable.

Degree	Name	Example
0	constant polynomial	5
1	linear polynomial	2x + 3
2	quadratic polynomial	x² − 5x + 6
3	cubic polynomial	x³ − 1

By number of terms: monomial (1), binomial (2), trinomial (3).

3. Value and zero of a polynomial

The value of p(x) at x = a is p(a) — substitute and evaluate.

A zero (root) of p(x) is a value of x for which p(x) = 0. For a linear polynomial ax + b:

$a x + b = 0 \Rightarrow x = - \frac{b}{a}$

So a linear polynomial has exactly one zero. A polynomial of degree n has at most n zeroes.

4. The Remainder Theorem

If a polynomial p(x) is divided by (x − a), the remainder is p(a).

This lets you find a remainder without long division — just evaluate p(a).

5. The Factor Theorem

(x − a) is a factor of p(x) if and only if p(a) = 0.

So to test whether (x − a) divides p(x), check if p(a) = 0. This is the key tool for factorising higher-degree polynomials: find one root by trial, then divide out its factor.

6. Worked examples

(a) Find the remainder when p(x) = x³ − 2x² + x + 1 is divided by (x − 1).

By the Remainder Theorem, remainder = p(1) = 1 − 2 + 1 + 1 = 1.

(b) Is (x − 2) a factor of p(x) = x³ − 3x² + 4?

p(2) = 8 − 12 + 4 = 0. Since p(2) = 0, by the Factor Theorem (x − 2) is a factor.

7. Quick recap

A polynomial has whole-number powers; know term, coefficient, constant, degree.
Types by degree: constant (0), linear (1), quadratic (2), cubic (3).
A zero makes p(x) = 0; a linear ax + b has the single zero x = −b/a; degree n → at most n zeroes.
Remainder Theorem: remainder on dividing by (x − a) is p(a).
Factor Theorem: (x − a) is a factor ⇔ p(a) = 0 — the engine of factorisation.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Polynomial

p(x) = aₙxⁿ + … + a₁x + a₀

Powers of x are whole numbers.

Zero of linear polynomial

ax + b = 0 ⇒ x = −b/a

Exactly one zero.

Remainder Theorem

remainder of p(x) ÷ (x − a) = p(a)

No division needed.

Factor Theorem

(x − a) is a factor ⇔ p(a) = 0

Test by substitution.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Calling √x or 1/x a polynomial term

✓ Powers must be whole numbers. x^(1/2) and x^(−1) disqualify the expression from being a polynomial.

WATCH OUT

✗ Confusing the degree with the number of terms

✓ Degree = highest power; number of terms gives monomial/binomial/trinomial. They are different ideas.

WATCH OUT

✗ Using (x + a) when checking p(a) in the Factor Theorem

✓ For factor (x − a) test p(a); for factor (x + a) test p(−a). Match the sign carefully.

WATCH OUT

✗ Thinking a quadratic always has two real zeroes

✓ Degree n means AT MOST n zeroes; a quadratic may have two, one (repeated) or none real.

WATCH OUT

✗ Sign slips when substituting negative values

✓ Bracket the value: p(−2) = (−2)³ − 3(−2)² … keep brackets to get the signs right.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

In-text (terms & degree)

Identify coefficients, constant, degree

Questions

In-text (value & zeroes)

Evaluate p(a); find zeroes of linear polynomials

Questions

In-text (theorems)

Remainder and Factor Theorem applications

Questions

End-of-chapter Ex

End-of-chapter exercise

Classification, theorems, factorisation

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Degree

State the degree of 5x³ − 2x² + 7.

▶ Show solution

The highest power of x is 3.

✦ Answer: degree 3 (cubic).

Q2EASY· Coefficient

What is the coefficient of x² in 4x³ − 6x² + x − 9?

▶ Show solution

The term in x² is −6x².

✦ Answer: −6.

Q3EASY· Zero

Find the zero of the polynomial 2x + 6.

▶ Show solution

Step 1 — 2x + 6 = 0 → x = −6/2.

Step 2 — x = −3.

✦ Answer: x = −3.

Q4MEDIUM· Value

If p(x) = x² − 3x + 2, find p(2) and p(−1).

▶ Show solution

Step 1 — p(2) = 4 − 6 + 2 = 0.

Step 2 — p(−1) = 1 + 3 + 2 = 6.

✦ Answer: p(2) = 0, p(−1) = 6.

Q5MEDIUM· Remainder Theorem

Find the remainder when x³ + 3x² + 3x + 1 is divided by (x + 1).

▶ Show solution

Step 1 — divisor (x + 1) → evaluate at x = −1.

Step 2 — p(−1) = −1 + 3 − 3 + 1 = 0.

✦ Answer: remainder = 0.

Q6MEDIUM· Factor Theorem

Show that (x − 1) is a factor of x³ − 1.

▶ Show solution

Step 1 — let p(x) = x³ − 1; test x = 1.

Step 2 — p(1) = 1 − 1 = 0.

Step 3 — since p(1) = 0, (x − 1) is a factor.

✦ Answer: p(1) = 0 ⇒ (x − 1) is a factor.

Q7HARD· Factorise

Factorise x³ − 6x² + 11x − 6 using the Factor Theorem.

▶ Show solution

Step 1 — try x = 1: 1 − 6 + 11 − 6 = 0, so (x − 1) is a factor.

Step 2 — divide to get x² − 5x + 6.

Step 3 — factorise: x² − 5x + 6 = (x − 2)(x − 3).

✦ Answer: (x − 1)(x − 2)(x − 3).

Q8HARD· Find constant

Find k if (x − 2) is a factor of x³ − kx² + 2x + 4.

▶ Show solution

Step 1 — (x − 2) is a factor → p(2) = 0.

Step 2 — p(2) = 8 − 4k + 4 + 4 = 16 − 4k = 0.

Step 3 — 4k = 16 → k = 4.

✦ Answer: k = 4.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Polynomial: whole-number powers; know term, coefficient, constant, degree.
•By degree: constant (0), linear (1), quadratic (2), cubic (3); by terms: mono/bi/trinomial.
•Value = p(a); zero makes p(x) = 0; linear ax + b has one zero x = −b/a; degree n → ≤ n zeroes.
•Remainder Theorem: remainder of p(x) ÷ (x − a) = p(a).
•Factor Theorem: (x − a) is a factor ⇔ p(a) = 0.
•Factorise a cubic by finding one zero (Factor Theorem), dividing, then splitting the quadratic.

Rajasthan (RBSE) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 5–7 marks

Question type	Marks each	Typical count	What it tests
MCQ / fill in the blank	1	1–2	Degree, coefficient, type of polynomial
Short answer	2	2	Value p(a), Remainder/Factor Theorem checks
Short/Long answer	3	1	Factorising a cubic; finding an unknown coefficient

Prep strategy

Learn the Remainder/Factor Theorems as a pair and when to use each
Practise factorising cubics: find one root, then split the quadratic
Always bracket negative substitutions to avoid sign errors
Classify by degree AND number of terms quickly

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Modelling curves

Polynomials describe trajectories, profit curves and growth — evaluating p(a) predicts an outcome.

Computer graphics

Bézier and spline curves used in fonts and animation are built from polynomials.

Engineering & error-correction

Polynomial roots and factorisation power signal processing and CRC checks.

Economics

Cost/revenue models are often polynomials; their zeroes mark break-even points.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

For a remainder, just compute p(a) — never do long division unless asked.
To test a factor, match the sign: (x − a) → p(a); (x + a) → p(−a).
Factorise cubics by trialling small integer roots (±1, ±2, factors of the constant).
Show substitution with brackets and state the theorem you used.
End factorisation answers fully factored into linear/irreducible parts.

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Relationship between zeroes and coefficients (Vieta's formulas).
Rational Root Theorem for spotting candidate zeroes quickly.
Polynomial division algorithm and synthetic division.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

RBSE Class 9 Annual (BSER Ajmer)High — a theorem/factorisation question every year

NTSE / NMMSMedium — polynomial-value MCQs

JEE FoundationVery high — base for Class 10 Polynomials and Quadratics

Maths Olympiad (IMO)Medium — factor theorem and roots

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

Yes. Class 9 (2026-27) uses the new NCF NCERT 'Ganita Prakash 9' book; 'Introduction to Linear Polynomials' covers polynomials, degree, zeroes and the Remainder/Factor Theorems. BSER Ajmer sets the RBSE paper.

The Remainder Theorem gives the remainder p(a) for division by (x − a). The Factor Theorem is the special case where that remainder is zero — then (x − a) is a factor.

By definition. Terms like x^(1/2) or x^(−1) are not polynomial terms, so an expression with them is not a polynomial.

No. A polynomial of degree n has at most n real zeroes.

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