NCERT Solutions for Class 10 Maths Chapter 2 Polynomials
NCERT Solutions for Class 10 Maths Chapter 2 Polynomials - Free PDF
NCERT Solutions for Class 10 Maths Chapter 2 Polynomials - Free PDF Download
You can opt for Chapter 2 - Polynomials NCERT Solutions for Class 10 Maths PDF for Upcoming Exams and also You can Find the Solutions of All the Maths Chapters below.
NCERT Solutions for Class 10 Maths
Polynomials
Chapter 2 Polynomials Class 10 is divided into four sections, three exercises and one optional exercise. The first section is with no exercise which talks about the constant, variables, coefficient and degree of polynomials in its introduction. The Second and Third section explains the geometrical representation of the zeros of polynomials and its relation with the coefficient of a polynomial. The most important topic of this chapter is the Division Algorithm which is discussed in the Fourth Section.
List of Exercises and Topics They Cover:
Exercise 2.1: Geometrical Meaning Of The Zeroes Of A Polynomial.
Exercise 2.2: Relationship Between Zeroes and Coefficient Of A Polynomial.
Exercise 2.3: Division Algorithm For Polynomials.
Exercise 2.4: Optional.
Variables - The Unknown Value
Have you ever wondered why children have different heights? Some children grow taller and some end up being shorter than average. Scientists have come closer to finding the answer by finding out the variables that are the cause of height.
The word ‘variable’ comes from the word ‘vary’ which means changing. Therefore, a variable can be any factor, trait or condition that can change by the differing amount or it is the unknown term whose value is not known. Example- Your height is dependent on the amount of protein- and nutrients you have. It is also dependent on your DNA that means if your parents are tall then there are more chances of you being tall whereas short parents usually have short kids. The height of the kid also depends on its rate of work or activities. Children with more activities like jumping, running, skipping etc tend to grow more. Thus, nutrients, DNA and activities are the three variables that control the height in our body. These variables keep changing from body to body.
To understand variables better, let us take one more example. While cooking dal, the quantity of water is thrice the quantity of lentils. That means if you add 1 cup lentils then you should add three cups of water. This process can be expressed as,
“3x + x”
Here, the quantity of lentil is variable. So if the quantity of lentils changes then the quantity of water should also change.
In a World Full of Variables, There is Always Something Constant.
There is one interesting thing about constants and that is this it never changes. In Maths, a constant is a value that is a fixed number on its own. For example - In the equation 9-x = 5, 9 and 5 are two constants whose values will not change whereas the value of x is not known. Thus, x is a variable.
Can Constant be a Coefficient too?
Since now we already know about variables, it is easier for us to understand the constant and coefficient. A coefficient is usually the number that is multiplied with the variable or letters. For example in ‘5x + y - 7’, 5 is a coefficient of x in the term 5x because it is a number which is multiplied to the unknown variable x. Also, in the term y, one can be considered as the coefficient of y because y can be written as 1xy.
We know that the coefficient is the number which is multiplied with the variables but Constants are terms without variables. Therefore a coefficient can never be constant and vice versa. In the aforementioned example, -7 is constant.
Like Terms:
Like terms are terms having the same variables raised to the same power. In 5x + y - 7, all the terms (i.e, 5x, y and -7) have different variables (or no variables). No common variable is found so there are no like terms.
In 5a + 2b - 3a + 4 the terms like 5a and -3a are like terms whereas 4 is constant..
What is a Polynomial?
The word Polynomial comes from the word In simple word poly (meaning "many") and nominal (meaning "term"). Therefore a polynomial is an expression consisting of many terms, where each term holds at least one variable. The variables can be elevated to power and further multiplied by a coefficient but the simplest polynomials hold one variable. The terms are separated by with + or - sign and the variables and numbers can be combined using addition, subtraction, multiplication or division but it can never be divided by a variable (i.e, a term can never be like 2/x). A polynomial can also not have infinite terms. It always has a finite sum of terms in which all variables have whole number exponents and no variable appears in a denominator. Also, in each term the numerical coefficient is always a real number.
Polynomials are Composed of the Following:
Constants such as 3, −20, or ½, etc.
Variables such as g, h, x, y, etc.
Exponents such as 2 in \[y^{2}\] or 5 in \[x^{5}\] etc
Examples of polynomials: 5\[x^{3}\] – 2\[x^{2}\] + x – 13 and \[x^{2}\]\[y^{3}\] + xy.
Degree of a Polynomial
It is simply the highest of the powers or exponents on the terms present in the algebraic expression.
Example: In 7x – 5, the first term is 7x, whereas the second term is -5. The power on the variable of first term 7x is 1 and on the second term -5, is 0. Since, the highest exponent is 1, the degree of 7x – 5 is also 1.
Types of Polynomials
Polynomials can be classified either on the basis of:
Number of terms.
Degree of polynomial.
Classification on the Basis of Terms
A polynomial can have one term, two term, three or more than three terms.
Monomials – ‘Mono’ stands for one and ‘mial’ stands for terms thus an algebraic expression with one term is called monomial. For example, 2x + 5x + 10x is a monomial because when we add the like terms it results in 17x. Furthermore, 4t, 21x2y, 9pq etc are also monomials because each of these expressions has only one term.
Binomials – ‘Bi’ stands for two and ‘mial’ stands for terms therefore an algebraic expression with two, unlike terms are called binomials. For example, 3x + 4x2 and 10pq + 13p2q are binomials.
Trinomials – ‘Tri’ stands for three and ‘mial’ stands for terms thus an algebraic expression with three unlike terms are called trinomials. For example- 3x + 5x2 – 6x3 and 12pq + 4x2 – 10 are trinomials.
Classification on the Basis of Degrees
The highest value of the exponent in the expression is known as Degree of Polynomial. The degree of a polynomial is the largest exponent. It is also known as an order of the polynomial. While finding the degree of the polynomial, the polynomial powers of the variables should be either in ascending or descending order.
Linear Polynomial: If the expression is of degree one then it is called a linear polynomial. For Example 5x+2,50z+3.
Quadratic Polynomial: If the expression is of degree two then it is called a quadratic polynomial.For Example 2\[x^{2}\] + 4x, 3\[x^{2}\] + 3x + 3.
Cubic Polynomial: If the expression is of degree three then it is called a cubic polynomial.For Example 2\[\x^{3}\] + 5x, 8\[\y^{3}\] + 6y + 3.
Zeros Of Polynomials
If the value of every coefficient of a variable is zero then it is called the zeros of a Polynomial.. In order to find the relationship between the zeroes and coefficients of a given quadratic polynomial, we can find the zeros of the polynomial by factorisation method that is, by taking the sum and product of these zeros.
Operations on Polynomial
There are four main polynomial operations which are:
Addition of Polynomials
Subtraction of Polynomials
Multiplication of Polynomials
Division of Polynomials
Visual Representation of Polynomial
Types of Polynomials:
Operations on Polynomials:
Difference between Arithmetic Long Division And Polynomial Long Division:
Polynomial long division is a method for dividing a polynomial by another polynomial of a lower degree.
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NCERT Solutions Class 10 Maths Chapter 2 Polynomials are provided here to help you in learning efficiently for their exams. The subject experts of Maths have prepared these solutions to help you prepare well for your exams. They solve these solutions in such a way that it becomes easier for you to practice the questions of Chapter 2 Polynomials using the Solutions of NCERT. They make it simple for you to learn by adding step-wise explanations to these Maths NCERT Class 10 Solutions.
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Why are Polynomials so Important?
Polynomials are considered to be an important part of the "language" of mathematics and algebra. It is used in nearly every field of Maths and algebra to express numbers as a result of mathematical operations. Polynomials are also called "building blocks" in other mathematical expressions, such as rational expressions.
Polynomials find its applications in many mathematical processes done in our everyday life that can be interpreted as polynomials. For example - Summing the cost of items on a grocery bill, calculating the distance traveled of a vehicle or object, calculating perimeter, area, and volume of geometric figures, etc all the above can be interpreted as polynomials.
Section 2.2 - Exercise 2.1
This exercise contains only one type of question and that is finding the number of zeros of a polynomial in the given graphs. There are a total of six graphs in this exercise.
Section 2.3 - Exercise 2.2
The second exercise of Chapter 2 polynomial contains two types of questions with 6 questions each.
Type 1: Finding the zeros of quadratic polynomials and verifying the relationship between the zeros and the coefficients.
Type 2: Finding the quadratic polynomial with sum and product of its zeros.
Section 2.4 - Exercise 2.3
Exercise 2.3 is considered to be the main exercise of the chapter Polynomial. Maximum questions from this chapter in Board Exams comes from this section. In section 2.4, we learn about the method of dividing one polynomial by another. The method of verification remains the same that is,
There are five types of questions found in this exercise:
Type 1: Division of polynomial by another polynomial
Type 2: Verification of the factor of polynomial
Type 3: Obtaining Zeros of a polynomial.
Type 4: Finding divisor of the polynomial by using dividend, quotient and remainder.
Type 5: Examples of polynomials satisfying the given division algorithm.
Summary
Polynomials of degrees 1, 2, 3 are called linear, quadratic and cubic polynomials respectively
A quadratic polynomial with real coefficient is always in the form of a\[x^{2}\] + bx + c where a, b and c are real numbers but cannot be equal to zero.
The zeroes of a polynomial are precisely the x-coordinates of the points, where the graph of the polynomial intersects the x-axis.
A quadratic polynomial can have maximum 2 zeroes and a cubic polynomial can have maximum 3 zeroes.
If x and y are the zeroes of the quadratic polynomial of form a\[x^{2}\] + bx + c then,
x+y = - b/a, xy = c/a
If x, y and z are the zeros of the cubic polynomial ax3+bx2+cx+d, then
x+y+z = -b/a
ax+by+cz = c/a
xyz = - d/a
Q1. What is the difference between remainder theorem and factor theorem?
Remainder theorem states that if a polynomial p(x) is divided by (x - a), then the remainder is obtained by evaluating the expression p(a).
While, Factor theorem states that (x - a) will be a factor of polynomial p(x) only if the remainder obtained by evaluating the expression p(a) equals zero i.e; p(a) = 0.
For example: let p(x) = x² - 6x + 9, then find the remainder when it is divided by (x - 1).
So, according to the remainder theorem, remainder is obtained by evaluating the expression p(1)
p(1) = 1² - 6(1) + 9
= 1 - 6 + 9 = 4.
Therefore, the remainder is 4 when p(x) = x² - 6x + 9 is divided by (x - 1).
Now, check whether (x - 3) is the factor of p(x) = x² - 6x + 9.
So, according to factor theorem (x - 3) is the factor of p(x) = x² - 6x + 9 only if p(3) = 0.
p(3) = 3² - 6(3) + 9
= 9 - 18 + 9 = 0.
Therefore, (x - 3) is the factor of p(x) = x² - 6x + 9.
Q2: What is the geometrical meaning of the zeroes of a Polynomial?
For a linear polynomial ax + b, a ≠ 0, the graph of y = ax + b is a straight line which intersects the x-axis at exactly one point at (-b/a , 0). Therefore, the geometrical meaning of the zeroes of linear polynomial ax + b, a ≠ 0 is the x-coordinate of the point where the graph of y = ax + b intersects the x-axis.
For a quadratic polynomial ax² + bx + c, a ≠ 0, the graph of the equation y = ax² + bx + c is parabola, which is ‘U’ shaped either open upwards or open downwards depending on whether a > 0 or a < 0. Therefore, the geometrical meaning of a quadratic polynomial is that it can have either two distinct zeroes or two equal zeroes (i.e., one zero), or no zero. This means that a polynomial of degree 2 has at most two zeroes.
In general, for any given polynomial p(x) of degree n, the graph of y = p(x) intersects the x-axis at atmost n points. Therefore, the geometrical meaning for a polynomial p(x) of degree n is that it has at most n zeroes.
Q3. What are the relationships between zeroes and coefficients of (a) Quadratic polynomial (b) Cubic polynomial?
For quadratic polynomial ax² + bx + c, a ≠ 0, if α and β are its zeroes, then relationship between the zeroes and coefficients of quadratic polynomial is:
Sum of zeroes = α + β = -b/a = -(coefficient of x) / coefficient of x²
Product of zeroes = αβ = c/a = constant term / coefficient of x².
For cubic polynomial ax³ + bx² + cx + d, a ≠ 0, if α, β and γ are its zeroes, then relationship between the zeroes and coefficients of cubic polynomial is:
Sum of zeroes = α + β + γ = -b/a = -(coefficient of x²) / coefficient of x³.
Sum of product of zeroes taken two at a time = αβ + βγ + γα = c/a = coefficient of x / coefficient of x³.
Product of zeroes = α β γ = d/a = constant term / coefficient of x³.
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