# NCERT Solutions for Class 10 Maths Chapter 2 – Polynomials

NCERT solutions for class 10 Maths Chapter 2 – Polynomials have been prepared by well-trained Maths experts to help you excel in CBSE Class 12 board exams. We understand that board exams are very crucial for you, so we have carefully compiled Polynomials to help you become thorough with all concepts covered in NCERT Solutions covers significant concepts such as the geometrical meaning of the zeros of a polynomial, relationship between zeros and coefficients of a polynomial, and division algorithm for polynomials.

Polynomials are based on the latest** **guidelines of CBSE and the given curriculum in CBSE NCERT Class 10 Maths book.** **This chapter contains a total of 4 exercises and 16 questions, all of which are solved with accuracy and clarity using the best of the solving methods.

Polynomials** **prepare you completely for your class 10^{th} board exams by providing ample practice. The logical flow of our NCERT solutions helps you learn faster, better and retain more. They are just what you need to excel in your board exams.

## NCERT Class 10 Maths Chapter 2 – Polynomials

**Concepts covered in NCERT CBSE Class 10 Maths Chapter 2 are discussed herein a crisp and condensed form.**

**Introduction**

This part of the chapter teaches you that a polynomial is an algebraic expression in which the exponent on any variable is a whole number. Further, it tells you that if p(x) is a polynomial in x, the highest power of x in p(x) is called the degree of the polynomial p(x). Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.

If p(x) is a polynomial in x, and if k is any real number, then the value obtained by replacing x by k in p(x), is called the value of p(x) at x = k and is denoted by p(k). Further, a real number k is said to be a zero of a polynomial, p(x), if p(k) = 0.

**Geometrical Meaning of the Zeroes of a Polynomial**

This portion of the chapter tells you that zero of a linear polynomial is related to its coefficients. Next, it tells you about geometrical representations of linear and quadratic polynomials and the geometrical meaning of their zeroes. The zeroes of a polynomial *p*(*x*) are precisely the x-coordinates of the points, where the graph of *y* = *p*(*x*) intersects the *x*-axis.

The graph of a linear polynomial is a straight line, and it intersects the *x*-axis at exactly one point.

A quadratic polynomial in *x* with real coefficients is of the form a*x*2 + b*x* + c, where a, b, c are real numbers with a ≠ 0. The graph of a quadratic polynomial is a parabola. It looks like ∩ opening upwards or downwards depending upon the polynomial expression. This parabola can cut *x*-axis at zero, one or two points (illustrated in the figure given below). A quadratic polynomial can have either two distinct zeros or two equal zeroes (i.e., one zero), or no zero. This also means that a polynomial of degree 2 has at most two zeros.

Given below is the geometrical meaning of the zeroes of a cubic polynomial. There are at most 3 zeroes for any cubic polynomial. In other words, any polynomial of degree 3 can have at most three zeroes.

In general, given a polynomial *p*(*x*) of degree *n*, the graph of *y* = *p*(*x*) intersects the *x*-axis at maximum *n* points. Therefore, a polynomial *p*(*x*) of degree *n* has at most *n* zeros.

To summarize –

A polynomial of degree n has at most n zeros.

- A linear polynomial has one zero,
- A quadratic polynomial has at most two zeros.
- A cubic polynomial has at most 3 zeros.

**Relationship between Zeroes and Coefficients of a Polynomial**

The information given in this chapter can be summarized as follows.

- Zero of a linear polynomial, a
*x*+ b is – (b/a). - If α and β are the zeroes of a quadratic polynomial a
*x*2 + b*x*+ c, then, α + β = -b/a

Sum of zeroes = – (coefficient of *x*) /coefficient of *x*2

αβ = c/a

Product of zeroes = constant term / coefficient of *x*2

- If α, β and γ are the zeroes of a cubic polynomial a
*x*3 + b*x*2 + c*x*+ d, then

α+β+γ = – (b/a)

αβ + βγ + γα = c/a

αβγ = – (d/a)

**Division Algorithm for Polynomials**

This part of the chapter tells you how to find the other two zeros of a cubic polynomial when only one of its zero out of three is given. This can be developing a division algorithm for the polynomial. One polynomial can be divided by the other following these steps –

Step 1-arrange the terms of the dividend and the divisor in the decreasing order of their degrees. Step 2 – Divide the highest degree term of the dividend by the highest degree term of the divisor to obtain the first term of the quotient, then carry out the division process. Step 3 – The remainder from the previous division becomes the dividend for the next step. You have to stop the division process when the remainder is zero, or its degree is less than the degree of the divisor.

If *p*(*x*) and *g*(*x*) are any two polynomials with *g*(*x*) ≠ 0, then we can find polynomials *q*(*x*) and *r*(*x*) such that *p*(*x*) = *g*(*x*) × *q*(*x*) + *r*(*x*), where *r*(*x*) = 0 or degree of *r*(*x*) < degree of *g*(*x*). This result is known as the Division Algorithm for polynomials.

### NCERT Class 10 Maths Chapter 2 – Polynomials Exercises

CBSE NCERT Class 10 Maths Chapter 2 consists of a total of 4 exercises, although the fourth exercise is optional, and the total number of questions is 16. NCERT solutions for class 10 Maths Chapter 2 – Polynomials provides the best answers to the above questions for your clear understanding.

**Exercise 2.1 **

The first exercise of NCERT Class 10 Maths Chapter 2 has only 1 question which needs you to find zeros of some polynomials, *p*(*x*).

**Exercise 2.2**

This exercise of NCERT CBSE Class 10 Maths Chapter 2 contains 2 questions each having 6 parts. In question 1 you have to verify the relationship between the zeroes and the coefficients of given quadratic polynomials. Question 2 asks you to find a quadratic polynomial using the provided information.

**Exercise 2.3**

The third exercise of Class 10 Maths Polynomials consists of 5 questions. Questions 1, 2 and 4 ask you to perform divisions of given polynomials. Question 3 asks you to find all the zero of a polynomial.

**Exercise 2.4**

This optional exercise given in chapter 2 of CBSE NCERT Class 10 Maths book contains 5 questions. All of the questions are based on verifying the relationship between the zeroes and the coefficients.

NCERT solutions for class 10 Maths Chapter 2 – Polynomials are written using the best available strategies to answer these questions and lets you grasp the content strongly.