Polynomials Class 9 ||Maths|| Chapter 2 Notes

Polynomials Class 9 ||Maths|| Chapter 2 Notes

2.1 Introduction

• A polynomial is an algebraic expression consisting of variables (also known as indeterminates), constants, and exponents combined using addition, subtraction, and multiplication.
• Example: $3x^2+2x-5$ is a polynomial.

2.2 Polynomials in One Variable

• A polynomial can have one or more variables. When it involves only one variable, it is called a polynomial in one variable.
• General form: $a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$, where $a_n,a_{n-1},\dots ,a_1,a_0$ are constants and $n$ is a non-negative integer (degree of the polynomial).

Examples:

• $3x^2+2x-5$ is a polynomial in $x$.
• $x^3-4x+7$ is another polynomial in one variable.

2.3 Degree of a Polynomial

• The degree of a polynomial is the highest power of the variable in the polynomial.

Examples:

• In $4x^3+2x^2-5$, the degree is 3.
• In $x^4-3x^2+x$, the degree is 4.

Types of Polynomials based on Degree:

1. Constant Polynomial: A polynomial of degree 0. Example: $5$.
2. Linear Polynomial: A polynomial of degree 1. Example: $2x+3$.
3. Quadratic Polynomial: A polynomial of degree 2. Example: $x^2-4x+7$.
4. Cubic Polynomial: A polynomial of degree 3. Example: $x^3+3x-1$.

2.4 Zeroes of a Polynomial

• A zero of a polynomial is a value of the variable that makes the polynomial equal to zero.

Example:

• For $p(x)=x^2-5x+6$, if $p(2)=0$ and $p(3)=0$, then 2 and 3 are the zeroes of the polynomial.

2.5 Remainder Theorem

• If a polynomial $p(x)$ is divided by $x-a$, then the remainder is $p(a)$.
• Statement: For a polynomial $p(x)$, when divided by $x-a$, the remainder is $p(a)$.

Example:

• For $p(x)=x^2-3x+4$, if we divide by $x-2$, the remainder is $p(2)=2^2-3(2)+4=4-6+4=2$.

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2.6 Factorization of Polynomials

• Factorization is the process of expressing a polynomial as a product of its factors.

Steps to Factor a Polynomial:

1. Find the common factor (if any).
2. Apply the splitting middle term method (for quadratic polynomials).
3. Use algebraic identities to simplify.

Example:

• Factor $x^2-5x+6$:
  • Find two numbers whose product is 6 and whose sum is -5.
  • Split the middle term: $x^2-2x-3x+6=x(x-2)-3(x-2)=(x-2)(x-3)$.

2.7 Algebraic Identities

• Algebraic identities are useful for expanding and factorizing expressions.

Important Identities:

1. $(a+b{)}^2=a^2+2ab+b^2$
2. $(a-b{)}^2=a^2-2ab+b^2$
3. $a^2-b^2=(a+b)(a-b)$
4. $(x+a)(x+b)=x^2+(a+b)x+ab$

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2.8 Division of Polynomials

• When dividing one polynomial by another, use the long division method.

Steps:

1. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
2. Multiply the entire divisor by this term and subtract it from the dividend.
3. Repeat the process with the new dividend until the degree of the remainder is less than the degree of the divisor.

Example:

• Divide $x^3-3x^2+5x-2$ by $x-1$:
  • Step 1: $\frac{x^3}{x}=x^2$.
  • Step 2: Multiply $x^2\times (x-1)=x^3-x^2$.
  • Step 3: Subtract: $(x^3-3x^2+5x-2)-(x^3-x^2)=-2x^2+5x-2$.
  • Continue until the remainder is of lower degree.

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Key Concepts

1. Degree of a Polynomial: The highest power of the variable in the polynomial.
2. Zeroes of a Polynomial: The values of $x$ that make the polynomial equal to zero.
3. Remainder Theorem: When a polynomial $p(x)$ is divided by $x-a$, the remainder is $p(a)$.
4. Factorization: Expressing a polynomial as a product of its factors.
5. Algebraic Identities: Equations that hold true for all values of the variables involved, used for simplification and factorization.

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Practice Problems

1. Find the zeroes of the polynomial $p(x)=x^2-7x+10$.
2. Divide $2x^3-9x^2+5x-1$ by $x-2$.
3. Factor the polynomial $x^2-16$ using the identity $a^2-b^2=(a+b)(a-b)$.
4. Verify the Remainder Theorem for $p(x)=x^2-3x+2$ when divided by $x-1$.

Tips for Study:

• Practice identifying degrees and coefficients of various polynomials.
• Work through exercises on addition, subtraction, and multiplication of polynomials.
• Factorize different types of polynomials to enhance your understanding.
• Familiarize yourself with the concept of zeroes and learn to solve simple polynomial equations.