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NCERT Solutions for Exercise 2.3 Class 10 Maths Chapter 2 - Polynomials

NCERT Solutions for Exercise 2.3 Class 10 Maths Chapter 2 - Polynomials

Edited By Ramraj Saini | Updated on Nov 08, 2023 06:31 PM IST | #CBSE Class 10th

NCERT Solutions For Class 10 Maths Chapter 2 Exercise 2.3 Polynomials

NCERT Solutions for class 10 maths ex 2.3 Polynomials is discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the latest syllabus and pattern of CBSE 2023-24. Exercise 2.3 of Class 10 Maths introduces the notion of polynomials in x with real coefficients, the notion of polynomials of degrees 1, 2, 3, quadratic equation, and many more with numerical problems on the Polynomials chapter.

This Story also Contains
  1. NCERT Solutions For Class 10 Maths Chapter 2 Exercise 2.3 Polynomials
  2. Download PDF of NCERT Solutions For Class 10 Maths Chapter 2 Exercise 2.3 Polynomials
  3. Access Exercise 2.3 Class 10 Maths Answers
  4. More About NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.3
  5. Key Features of NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.3
  6. NCERT Solutions of Class 10 Subject Wise
  7. Subject Wise NCERT Exemplar Solutions
NCERT Solutions for Exercise 2.3 Class 10 Maths Chapter 2 - Polynomials
NCERT Solutions for Exercise 2.3 Class 10 Maths Chapter 2 - Polynomials

The Class 10th Maths chapter 2 exercise 2.3 moves towards the topics like the class of addition, subtraction and multiplication of polynomials. Also, zeroes/roots of a polynomial function f(x). Also students can find all exercise listed below, practice them to command the maths concepts.

Download PDF of NCERT Solutions For Class 10 Maths Chapter 2 Exercise 2.3 Polynomials

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Access Exercise 2.3 Class 10 Maths Answers

Polynomials Class 10 Maths Chapter 2 Excercise: 2.3

Q1 (1) Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder
in each of the following :
(i) p(x) = x^3 - 3x^2 + 5x - 3, g(x) = x^2 - 2

Answer: The polynomial division is carried out as follows

1635918804777

The quotient is x-3 and the remainder is 7x-9

Q1 (2) Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following : p(x) = x^4 - 3x^2 + 4x + 5, g(x) = x^2 + 1 - x

Answer:

The division is carried out as follows

1635919021607

The quotient is x^2+x-3

and the remainder is 8

Q1 (3) Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following :
p(x) = x^4 - 5x + 6, g(x) = 2 - x^2

Answer: The polynomial is divided as follows

1635919038492

The quotient is -x^2-2 and the remainder is -5x+10

Q2 (1) Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

t^2 - 3, 2t^4 + 3t^3 - 2t^2 - 9t - 12

Answer:

1635919067387

After dividing we got the remainder as zero. So t^2 - 3 is a factor of 2t^4 + 3t^3 - 2t^2 - 9t - 12

Q2 (2) Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

x^2 + 3x + 1, 3x^4 + 5x^3 - 7x^2 + 2x + 2

Answer: To check whether the first polynomial is a factor of the second polynomial we have to get the remainder as zero after the division

1635919079452

After division, the remainder is zero thus x^2+3x+1 is a factor of 3x^4 + 5x^3 - 7x^2 + 2x + 2

Q2 (3) Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial: x^3 - 3x + 1, x^5 - 4x^3 + x^2 + 3x + 1

Answer: The polynomial division is carried out as follows

1635919099878

The remainder is not zero, there for the first polynomial is not a factor of the second polynomial

Q3 Obtain all other zeroes of 3x^4 + 6x^3 - 2x^2 - 10x - 5 , if two of its zeroes are \sqrt {\frac5{3}} \: \:and \: \: - \sqrt {\frac{5}{3}}

Answer:

Two of the zeroes of the given polynomial are \sqrt {\frac5{3}} \: \:and \: \: - \sqrt {\frac{5}{3}} .

Therefore two of the factors of the given polynomial are x-\sqrt{\frac{5}{3}} and x+\sqrt{\frac{5}{3}}

(x+\sqrt{\frac{5}{3}})\times (x-\sqrt{\frac{5}{3}})=x^{2}-\frac{5}{3}

x^{2}-\frac{5}{3} is a factor of the given polynomial.

To find the other factors we divide the given polynomial with 3\times (x^{2}-\frac{5}{3})=3x^{2}-5


1635919117816

The quotient we have obtained after performing the division is x^{2}+2x+1

\\x^{2}+2x+1\\ =x^{2}+x+x+1\\ =x(x+1)+(x+1)\\ =(x+1)^{2}

(x+1) 2 = 0

x = -1

The other two zeroes of the given polynomial are -1.

Q4 On dividing x^3 - 3x^2 + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and –2x + 4, respectively. Find g(x).

Answer: Quotient = x-2

remainder =-2x+4

\\dividend=divisor\timesquotient+remainder\\x^3-3x^2+x+2=g(x)(x-2)+(-2x+4)\\x^3-3x^2+x+2-(-2x+4)=g(x)(x-2)\\x^3-3x^2+3x-2=g(x)(x-2)\\

g(x)=\frac{x^3-3x^2+3x-2}{x-2}

Carrying out the polynomial division as follows

1635919149497

g(x)={x^2-x+1}

Q5 (1) Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm
and
(i) deg p(x) = deg q(x)

Answer: deg p(x) will be equal to the degree of q(x) if the divisor is a constant. For example

\\p(x)=2x^2-2x+8\\q(x)=x^2-x+4\\g(x)=2\\r(x)=0

Q5 (2) Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and deg q(x) = deg r(x)

Answer: Example for a polynomial with deg q(x) = deg r(x) is given below

\\p(x)=x^3+x^2+x+1\g(x)=x^2-1\\q(x)=x+1\\r(x)=2x+2

Q 5 (3) Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and deg r(x) = 0

Answer:

example for the polynomial which satisfies the division algorithm with r(x)=0 is given below

\\p(x)=x^3+3x^2+3x+5\\q(x)=x^2+2x+1\\g(x)=x+1\\r(x)=4

More About NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.3

The problems from the concepts of division of polynomial p(x) with polynomial g(x) are covered in exercise 2.3 Class 10 Maths. The Introductory numerical questions of NCERT solutions for Class 10 Maths chapter 2 exercise 2.3 is to represent the numerical problems of finding quotient and remainder. And later on questions of Class 10 Maths chapter 2 exercise 2.3 is to find the polynomial if Quotient and remainder is given.

Also Read| Polynomials Class 10 Notes

Key Features of NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.3

  • The solved example before the exercise 2.3 Class 10 Maths and the NCERT solutions for Class 10 Maths chapter 2 exercise 2.3 are important as it covers questions from the basic division of Polynomials

  • If students can solve each and every question of this exercise 2.3 Class 10 Maths he will be able to grasp the factorization concept of the polynomial as given in Class 10th Maths chapter 2 exercise 2.4

  • For Class 10 final exams students may get a short answer or long answer questions from the type covered in the Class 10 Maths chapter 2 exercise 2.3

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Frequently Asked Question (FAQs)

1. Define Polynomial?

Polynomial '' comes from the word ‘Poly’ (Meaning Many) and ‘nomial’ (in this case meaning Term)-so it means many terms.

2. Can a in polynomial ax² + bx + c be equal to zero?

No, a ≠ 0 because it will turn into a linear equation

3. What happens when zeroes of quadratic equation are equal?

 When ax² + bx + c =0 has roots/zeros  α and β if they are equal, then sign of  a (coef of  x²) and c (constant term) must be equal

4. What is the formula of sum of roots of Polynomial ax² + bx + c =0?

sum of roots = α + β = -(b/a).

5. What is the formula of Product of roots of Polynomial ax² + bx + c =0?

 Product of roots = α * β = (c/a)

6. What does a polynomial with degree = 3 called?

A polynomial with degree  = 3 is called Cubic  polynomial

7. What is division algorithm in Polynomial?

Ans Consider the functions,  f(x) and g(x) are any two polynomials with for any value of x, g(x) ≠ 0, then we have derived an expression-
         f(x) = g(x) × q(x) + r(x)
         Dividend = Divisor * Quotient + Remainder

8. What happens when a polynomial is multiplied by itself?

Each term of the first polynomial is multiplied by each term of the second polynomial when the two polynomials are multiplied.

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