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POLYNOMIALS — Comprehensive AI Notes for Class 9 CBSE

1. Quick Overview

This chapter introduces polynomials — special types of algebraic expressions where variables have only non-negative whole number exponents. You'll learn how to identify polynomials, find their zeroes (values that make them equal to zero), factor them using techniques like the Factor Theorem, and apply powerful algebraic identities to simplify and expand expressions. By the end, you'll have tools to solve real problems in geometry, physics, and algebra — and these ideas form the foundation for everything in higher mathematics.

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2. Key Concepts (in Depth)

What is a Polynomial?

A polynomial in one variable is an algebraic expression where the variable appears only with non-negative whole number exponents, and all coefficients are real numbers. Think of it as a "sum of power terms."

For example, 5x³ – 2x² + 3x – 7 is a polynomial because x appears as x³, x², x¹, and x⁰ (the constant 7 = 7x⁰). But x + 1/x is not a polynomial because the second term has x⁻¹, a negative exponent.

The key reason polynomials are special: they behave nicely under addition, subtraction, and multiplication. You can always add two polynomials and get another polynomial. This closure property makes them fundamental in algebra.

Why it matters: Polynomials model real situations. If a square has side length x metres, its perimeter is 4x (a linear polynomial) and its area is x² (a quadratic polynomial). Engineers and scientists use polynomials to describe motion, electricity, and growth.

Degree and Types of Polynomials

The degree of a polynomial is the highest power of the variable that appears. In 3x⁵ – 2x + 4, the degree is 5 (because x⁵ is the highest power).

The student should understand these classifications:

• Linear polynomial (degree 1): e.g., 2x + 5. These model constant rates of change.
• Quadratic polynomial (degree 2): e.g., x² – 3x + 2. These appear in parabolic motion and optimisation problems.
• Cubic polynomial (degree 3): e.g., x³ – x. These are the first polynomials that can have up to 3 zeroes.

The general form is: p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where aₙ ≠ 0 and n is the degree.

A constant polynomial like 7 has degree 0 (because 7 = 7x⁰). The zero polynomial (0 itself) has no defined degree — it's a special case you must remember.

Why it matters: Degree tells you about the polynomial's behaviour. A degree-n polynomial has at most n zeroes, and this limit helps you understand solutions to equations.

Zeroes of a Polynomial

A zero of a polynomial p(x) is any real number c such that p(c) = 0. In other words, when you substitute c for x, the entire expression evaluates to zero.

Example: For p(x) = x – 3, when x = 3, we get p(3) = 3 – 3 = 0. So 3 is a zero.

Finding zeroes means solving an equation: if p(x) = 0, you're looking for the values of x that satisfy this. For a linear polynomial ax + b (where a ≠ 0), there is exactly one zero: x = –b/a.

For quadratic polynomials, there can be 0, 1, or 2 zeroes (depending on the discriminant, which you'll study later). A cubic polynomial can have up to 3 zeroes.

Why it matters: Zeroes tell you where a graph crosses the x-axis. They're crucial for solving real-world problems: "When does profit become zero?" or "At what distance is the object at ground level?"

Remainder Theorem and Factor Theorem

The Remainder Theorem states: When a polynomial p(x) is divided by (x – a), the remainder is p(a).

This is powerful: instead of doing long division, you simply evaluate p(a). For example, dividing x³ + 2x² – 5x + 3 by (x – 2) gives remainder p(2) = 8 + 8 – 10 + 3 = 9.

The Factor Theorem is the next logical step: (x – a) is a factor of p(x) if and only if p(a) = 0.

In other words, (x – a) divides p(x) evenly (no remainder) precisely when a is a zero.

Why it matters: This theorem is your key to factorising polynomials without tedious trial-and-error. If you can find one zero, you find one factor, then divide to reduce the problem.

Factorisation Strategies

For quadratic polynomials ax² + bx + c:

• Splitting the middle term: Rewrite bx as two terms whose product with a gives ac. For example, 6x² + 17x + 5: find two numbers that add to 17 and multiply to 30. These are 2 and 15. Then 6x² + 2x + 15x + 5 = 2x(3x + 1) + 5(3x + 1) = (3x + 1)(2x + 5).
• Using the Factor Theorem: Test small integer factors of the constant term until you find a zero.

For cubic polynomials: You must find at least one zero (by testing factors of the constant term), then use that to get one linear factor. Dividing gives a quadratic, which you can factor using the methods above.

Why it matters: Factorisation converts hard problems into easy ones. Solving x³ – 6x² + 11x – 6 = 0 is tedious until you factor it as (x – 1)(x – 2)(x – 3) = 0, giving zeroes 1, 2, 3 instantly.

Algebraic Identities

An algebraic identity is an equation true for all values of the variables. Unlike an equation (which we solve), an identity is universally true.

Key identities the student must memorise and understand:

1. (x + y)² = x² + 2xy + y² — expands a squared binomial.
2. (x – y)² = x² – 2xy + y² — same idea with subtraction.
3. x² – y² = (x + y)(x – y) — difference of squares factorises instantly.
4. (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx — squared trinomial has three square terms and three product terms.
5. (x + y)³ = x³ + y³ + 3xy(x + y) — cubed binomial.
6. (x – y)³ = x³ – y³ – 3xy(x – y) — same with subtraction.
7. x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – yz – zx) — factorises a sum of three cubes minus three times their product.

Why it matters: These identities save enormous time in two ways:

• Expanding: Instead of multiplying (3x + 4)(3x – 5) by hand, use identity 4 directly.
• Factorising: Recognise patterns. If you see x² + 6x + 9, spot it as (x + 3)² using identity 1 backwards.
• Computation: Evaluate 105² quickly as (100 + 5)² = 10000 + 1000 + 25 = 11025 without a calculator.

The student should spend time understanding why these work (by expanding), not just memorising them.

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3. Important Definitions

• Polynomial in one variable — An algebraic expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where all exponents are non-negative whole numbers and all coefficients are real.

• Coefficient — The numerical multiplier of a power of the variable. In 5x³ – 2x² + 3x – 7, the coefficient of x³ is 5, and the coefficient of x⁰ is –7.

• Term — A single component of a polynomial, like 5x³ or –7. Each term is a coefficient times a power of the variable.

• Degree of a polynomial — The highest exponent of the variable that appears with a non-zero coefficient. In 4x⁵ – x + 2, the degree is 5.

• Monomial — A polynomial with exactly one term, like 3x² or –5.

• Binomial — A polynomial with exactly two terms, like x + 1 or 2x² – 3.

• Trinomial — A polynomial with exactly three terms, like x² + 2x + 1.

• Linear polynomial — A polynomial of degree 1; general form ax + b with a ≠ 0.

• Quadratic polynomial — A polynomial of degree 2; general form ax² + bx + c with a ≠ 0.

• Cubic polynomial — A polynomial of degree 3; general form ax³ + bx² + cx + d with a ≠ 0.

• Zero of a polynomial — A real number c such that p(c) = 0. Also called a root of the equation p(x) = 0.

• Factor — A polynomial that divides another polynomial evenly (with remainder 0). If (x – a) is a factor of p(x), then p(x) = (x – a) · q(x) for some polynomial q(x).

• Algebraic identity — An equation involving variables that is true for all values of those variables.

• Remainder Theorem — When p(x) is divided by (x – a), the remainder equals p(a).

• Factor Theorem — (x – a) is a factor of p(x) if and only if p(a) = 0.

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4. Formulae / Rules / Laws

General Polynomial Form

Formula: p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + aₙ₋₂xⁿ⁻² + ... + a₁x + a₀

Variable meanings:

• aₙ, aₙ₋₁, ..., a₀ are coefficients (real numbers)
• aₙ ≠ 0 (ensures degree is exactly n)
• n is a non-negative integer (the degree)
• x is the variable

Worked example: Write the polynomial p(x) = 2x⁴ – 3x² + 5 in general form and identify coefficients.

Solution: p(x) = 2x⁴ + 0x³ – 3x² + 0x + 5

• a₄ = 2, a₃ = 0, a₂ = –3, a₁ = 0, a₀ = 5
• Degree = 4
• Terms: 2x⁴, –3x², 5

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Linear Polynomial Zero

Formula: If p(x) = ax + b (where a ≠ 0), the zero is x = –b/a

Variable meanings:

• a, b are constants with a ≠ 0
• The zero is the solution to ax + b = 0

Worked example: Find the zero of p(x) = 3x – 12.

Solution:
Set p(x) = 0: 3x – 12 = 0
3x = 12
x = 4

Check: p(4) = 3(4) – 12 = 12 – 12 = 0 ✓

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Identity I: Square of a Binomial

Formula: (x + y)² = x² + 2xy + y²

Variable meanings: x, y are any real numbers or algebraic expressions

Worked example: Expand (2a + 3b)².

Solution:
Let x = 2a, y = 3b.
(2a + 3b)² = (2a)² + 2(2a)(3b) + (3b)²
= 4a² + 12ab + 9b²

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Identity III: Difference of Squares

Formula: x² – y² = (x + y)(x – y)

Variable meanings: x, y are any real numbers or algebraic expressions

Worked example: Factorise 16p² – 25q².

Solution:
Rewrite: 16p² – 25q² = (4p)² – (5q)²
Apply identity: (4p + 5q)(4p – 5q)

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Identity V: Square of a Trinomial

Formula: (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx

Variable meanings: x, y, z are any real numbers or algebraic expressions

Worked example: Expand (a + 2b + 3c)².

Solution:
(a + 2b + 3c)² = a² + (2b)² + (3c)² + 2(a)(2b) + 2(2b)(3c) + 2(3c)(a)
= a² + 4b² + 9c² + 4ab + 12bc + 6ac

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Identity VI: Cube of a Binomial (Addition)

Formula: (x + y)³ = x³ + y³ + 3xy(x + y)

Alternative form: (x + y)³ = x³ + 3x²y + 3xy² + y³

Variable meanings: x, y are any real numbers or algebraic expressions

Worked example: Expand (2m + n)³.

Solution:
(2m + n)³ = (2m)³ + (n)³ + 3(2m)(n)(2m + n)
= 8m³ + n³ + 6mn(2m + n)
= 8m³ + n³ + 12m²n + 6mn²

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Identity VII: Cube of a Binomial (Subtraction)

Formula: (x – y)³ = x³ – y³ – 3xy(x – y)

Alternative form: (x – y)³ = x³ – 3x²y + 3xy² – y³

Variable meanings: x, y are any real numbers or algebraic expressions

Worked example: Expand (3p – 2q)³.

Solution:
(3p – 2q)³ = (3p)³ – (2q)³ – 3(3p)(2q)(3p – 2q)
= 27p³ – 8q³ – 18pq(3p – 2q)
= 27p³ – 8q³ – 54p²q + 36pq²

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Identity VIII: Sum of Three Cubes

Formula: x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – yz – zx)

Variable meanings: x, y, z are any real numbers or algebraic expressions

Worked example: Factorise 8a³ + 27b³ + 64c³ – 72abc.

Solution:
Rewrite: 8a³ + 27b³ + 64c³