i want the notes of maths polynomials
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Step-by-step explanation:
Polynomial derived from the words “poly” which means “many” and the word “nomial” which means “term”. In maths, a polynomial expression consists of variables which are also known as indeterminates and coefficients. The coefficients involve the operations of subtraction, addition, non-negative integer exponents of variables and multiplication. A detailed polynomials Class 9 notes are provided here along with some important questions so that students can understand the concept easily.
Polynomials Class 9 Topics
The topics and subtopics covered in class 9 polynomials chapter 2 include:
Introduction
Polynomials in One Variable
Zeros of Polynomials
Remainder Theorem
Factorisation of Polynomials
Algebraic Identities
Polynomial Definition
Polynomials are expressions with one or more terms with a non-zero coefficient. A polynomial can have more than one term. In the polynomial, each expression in it is called a term. Suppose x2 + 5x + 2 is polynomial, then the expressions x2, 5x, and 2 are the terms of the polynomial. Each term of the polynomial has a coefficient. For example, if 2x + 1 is the polynomial, then the coefficient of x is 2.
The real numbers can also be expressed as polynomials. Like 3, 6, 7, are also polynomials without any variables. These are called constant polynomials. The constant polynomial 0 is called zero polynomial. The exponent of the polynomial should be a whole number. For example, x-2 + 5x + 2, cannot be considered as a polynomial, since the exponent of x is -2, which is not a whole number.
The highest power of the polynomial is called the degree of the polynomial. For example, in x3 + y3 + 3xy(x + y), the degree of the polynomial is 3. For a non zero constant polynomial, the degree is zero. Apart from these, there are other types of polynomials such as:
Linear polynomial – of degree one
Quadratic Polynomial- of degree two
Cubic Polynomial – of degree three
This topic has been widely discussed in class 9 and class 10.
Example of polynomials are:
20
x + y
7a + b + 8
w + x + y + z
x2 + x + 1
Quadratic Equation Algebraic Identities
Quadratic Formula & Quadratic Polynomial Degree Of A Polynomial
For More Information On Quadratic Polynomial, Watch The Below Video.
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Polynomials in One Variable
Polynomials in one variable are the expressions which consist of only one type of variable in the entire expression.
Example of polynomials in one variable:
3a
2x2 + 5x + 15
Polynomial Class 9 Notes
To prepare for class 9 exams, students will require notes to study. These notes are of great help when they have to revise chapter 2 polynomials before the exam. The note here provides a brief of the chapter so that students find it easy to have a glance at once. The key points covered in the chapter have been noted. Go through the points and solve problems based on them.
Some important points in Polynomials Class 9 Chapter 2 are given below:
An algebraic expression p(x) = a0xn + a1xn-1 + a2xn-2 + … an is a polynomial where a0, a1, ………. an are real numbers and n is non-negative integer.
A term is either a variable or a single number or it can be a combination of variable and numbers.
The degree of the polynomial is the highest power of the variable in a polynomial.
A polynomial of degree 1 is called as a linear polynomial.
A polynomial of degree 2 is called a quadratic polynomial.
A polynomial of degree 3 is called a cubic polynomial.
A polynomial of 1 term is called a monomial.
A polynomial of 2 terms is called binomial.
A polynomial of 3 terms is called a trinomial.
A real number ‘a’ is a zero of a polynomial p(x) if p(a) = 0, where a is also known as root of the equation p(x) = 0.
A linear polynomial in one variable has a unique zero, a polynomial of a non-zero constant has no zero, and each real number is a zero of the zero polynomial.
Remainder Theorem: If p(x) is any polynomial having degree greater than or equal to 1 and if it is divided by the linear polynomial x – a, then the remainder is p(a).
Factor Theorem : x – c is a factor of the polynomial p(x), if p(c) = 0. Also, if x – c is a factor of p(x), then p(c) = 0.
The degree of the zero polynomial is not defined.
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
(x + y)3 = x3 + y3 + 3xy(x + y)
(x – y)3 = x3 – y3 – 3xy(x – y)
x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx)
Polynomials Class 9 Examples
Example 1:
Write the coefficients of x in each of the following:
3x + 1
23x2 – 5x + 1
Solution:
In 3x + 1, the coefficient of x is 3.
In 23x2 – 5x + 1, the coefficient of x is -5.
Example 2:
What are the degrees of following polynomials?
3a2 + a – 1
32x3 + x – 1
Solution:
3a2 + a – 1 : The degree is 2
32x3 + x – 1 : The degree is 3
Polynomials Class 9 Important Questions
Find value of polynomial 2x2 + 5x + 1 at x = 3.
Check whether x = -1/6 is zero of the polynomial p(a) = 6a + 1.
Divide 3a2 + x – 1 by a + 1.
Find value of k, if (a – 1) is factor of p(a) = ka2 – 3a + k.
Factorise each of the following:
4x2 + 9y2 + 16z2 + 12xy – 24yx – 16xz
2x2 + y2 + 8z2 – 2√2xy + 4√2yz – 8xz
The constant plynomial f(x) = 0 is called zero polynomial. Degree of zero polynomial is not defined. A polynomial of degree one is called a linear polynomil e.g. ax + b, where a ≠ 0. A polynomial of degree two is called a quadratic polynomial e.g. ax2 + bx + c where a ≠ 0.