CBSE Class 9 Maths Notes Chapter 2 Polynomials

Polynomials in one variable are algebraic expressions that involve only one variable, usually represented by $x$ . These expressions can have multiple terms, with each term consisting of a constant multiplied by a variable raised to a non-negative integer exponent. For example, 3𝑥2−5𝑥+7 3 x 2 − 5 x + 7 is a polynomial in one variable ( $x$ ). The highest power of the variable in the polynomial is called its degree. In the case of 3𝑥2−5𝑥+7 3 x 2 − 5 x + 7 , the degree is 2 because the highest power of $x$ is 2. Polynomials in one variable are fundamental in algebra and are used in various mathematical applications, such as solving equations, graphing functions, and modeling real-world scenarios.

Coefficient

In a polynomial expression like $2x+1$ , each term consists of a coefficient multiplied by a variable raised to a certain power. In this case, the term $2x$ has a coefficient of 2, which is the number multiplied by the variable $x$ . Similarly, the constant term $1$ can be considered as 1𝑥0 1 x 0 , where the coefficient of 𝑥0 x 0 (which is just 1) is also considered.

Types of Polynomial

Polynomials can be classified into different types based on various criteria, such as the number of terms they have or the highest power of the variable in the expression. Here are some common types of polynomials:

1. Monomial : A polynomial with only one term. For example, $2x$ is a monomial because it has only one term.
2. Binomial : A polynomial with two terms. For example, $5x+2$ is a binomial because it has two terms.
3. Trinomial : A polynomial with three terms. For example, $2x+5y-4$ is a trinomial because it has three terms.

The degree of a polynomial is determined by the highest power of the variable (or variables) present in the polynomial expression.

For example:

• In the polynomial 3𝑥2+5𝑥−1 3 x 2 + 5 x − 1 , the highest power of the variable $x$ is $2$ , so the degree of the polynomial is $2$ .
• In the polynomial 2𝑦3−𝑦+4 2 y 3 − y + 4 , the highest power of the variable $y$ is $3$ , so the degree of the polynomial is $3$ .
• In the polynomial 4𝑥4𝑦2−3𝑥𝑦+7 4 x 4 y 2 − 3 x y + 7 , the highest combined power of the variables $x$ and $y$ is $6$ (since $x$ has a power of $4$ and $y$ has a power of $2$ ), so the degree of the polynomial is $6$ .

The degree of a polynomial helps classify it and understand its behavior when performing mathematical operations like addition, subtraction, multiplication, and division. It's an important concept in algebra and polynomial arithmetic.

Algebraic Identities

Algebraic identities are algebraic equations which are valid for all values. The important algebraic identities used in Class 9 Maths chapter 2 polynomials are listed below:

• (x + y + z) ² = x ² + y ² + z ² + 2xy + 2yz + 2zx
• (x + y) ³ = x ³ + y ³ + 3xy(x + y)
• (x – y) ³ = x ³ – y ³ – 3xy(x – y)
• x ³ + y ³ + z ³ – 3xyz = (x + y + z) (x ² + y ² + z ² – xy – yz – zx)

Zeroes of Polynomial

The zeroes of a polynomial are the values of the variable that make the polynomial equal to zero when substituted into it. In other words, if $P(x)$ is a polynomial, then any value $a$ for which $P(a)=0$ is considered a zero (or root) of the polynomial. For example, consider the polynomial 𝑃(𝑥)=𝑥2−4 P ( x ) = x 2 − 4 . To find its zeroes, we set $P(x)$ equal to zero and solve for $x$ : 𝑥2−4=0 x 2 − 4 = 0 This equation can be factorized as $(x-2)(x+2)=0$ . So, the zeroes of the polynomial are $x=2$ and $x=-2$ . In general, a polynomial of degree $n$ can have at most $n$ zeroes. These zeroes may be real or complex numbers. The Fundamental Theorem of Algebra states that every polynomial equation of degree $n$ has exactly $n$ complex roots (including repeated roots). The zeroes of a polynomial are important in various mathematical contexts, such as solving equations, graphing functions, and understanding the behavior of polynomial functions.

Remainder Theorem

The Remainder Theorem is a fundamental concept in algebra that relates to polynomial division. It states that if a polynomial $P(x)$ is divided by a linear polynomial of the form $x-a$ , then the remainder is equal to $P(a)$ , where $a$ is any real number. In simpler terms, if you divide a polynomial by $x-a$ , the remainder you get will be the value of the polynomial evaluated at $a$ . For example, let's say we have the polynomial 𝑃(𝑥)=𝑥2+3𝑥−4 P ( x ) = x 2 + 3 x − 4 and we want to divide it by $x-2$ . According to the Remainder Theorem, the remainder will be $P(2)$ , which means we substitute $x=2$ into the polynomial $P(x)$ . So, 𝑃(2)=(2)2+3(2)−4=4+6−4=6 P ( 2 ) = ( 2 ) 2 + 3 ( 2 ) − 4 = 4 + 6 − 4 = 6 . Hence, when $P(x)$ is divided by $x-2$ , the remainder is $6$ . The Remainder Theorem is useful in various mathematical applications, including finding roots of polynomials, evaluating polynomial functions, and proving divisibility properties.

Factorisation of Polynomials

Factorization of polynomials involves expressing a given polynomial as the product of two or more simpler polynomials. For example, consider the polynomial 𝑥2−𝑥−6 x 2 − x − 6 . To factorize it, we look for two numbers whose product is $-6$ and whose sum is $-1$ , because the middle term of the polynomial is $x$ and the constant term is $-6$ . These numbers are $-3$ and $2$ , because $(-3)\times 2=-6$ and $(-3)+2=-1$ . Therefore, we can express 𝑥2−𝑥−6 x 2 − x − 6 as $(x-3)(x+2)$ by using these factors. This process of factorization helps simplify polynomial expressions and is a fundamental concept in algebra. It allows us to understand the structure of polynomials better and to solve various mathematical problems more efficiently.

Benefits of CBSE Class 9 Maths Notes Chapter 2 Polynomials

• Concept Clarity : These notes provide a clear explanation of polynomial concepts, ensuring that students understand the fundamentals thoroughly.
• Structured Learning : The notes are organized in a structured manner, covering topics sequentially. This helps students to follow a logical progression in their learning.
• Comprehensive Coverage : The notes cover all the essential topics related to polynomials, including definitions, types, factorization, remainder theorem, zeroes, and more. This comprehensive coverage ensures that students have a complete understanding of the chapter.
• Example Problems : The notes include solved examples that illustrate how to apply polynomial concepts in different scenarios. These examples help students grasp the application of theory in practical problems.
• Practice Questions : Along with solved examples, the notes also provide practice questions at the end of each topic or chapter. These questions allow students to test their understanding and reinforce their learning.
• Exam Preparation : By studying these notes, students can effectively prepare for exams. The clear explanations, solved examples, and practice questions help them revise the chapter thoroughly and build confidence for exams.

CBSE Maths Notes For Class 9

Chapter 1 – Number System

Chapter 2 – Polynomials

Chapter 3 – Coordinate Geometry

Chapter 4 – Linear Equations in Two Variables

Chapter 5 – Introduction to Euclid’s Geometry

Chapter 6 – Lines and Angles

Chapter 7 – Triangles

Chapter 8 – Quadrilaterals

Chapter 9 - Areas of Parallelograms and Triangles

CBSE Class 9 Maths Notes Chapter 10 - Circles Notes

CBSE Class 9 Maths Notes Chapter 11 - Constructions Notes

CBSE Class 9 Maths Notes Chapter 12 - Herons Formula Notes

CBSE Class 9 Maths Notes Chapter 13 - Surface Areas and Volumes Notes

CBSE Class 9 Maths Notes Chapter 14 - Statistics Notes

CBSE Class 9 Maths Notes Chapter 15 - Probability Notes

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