NCERT Solutions for Class 10 Maths Exercise 2.3 Chapter 2 Polynomials, which are provided here, are prepared by our subject experts. These solutions are created and later reviewed by the NCERT subject experts to make it easy for the students to grasp the concept easily. This ensures that these NCERT Solutions are easily understandable by the Class 10 Maths students.
The Division Algorithm for polynomials is explained in the Solutions of Chapter 2 Polynomials Class 10 Maths NCERT Exercise 2.3. These NCERT Class 10 solutions are prepared based on the NCERT guidelines. It ensures the whole syllabus is covered while providing the solutions so that the student shall not miss out on any concept.
Download the PDF of NCERT Solutions for Class 10 Maths Chapter 2 – Polynomials Exercise 2.3
Access answers of Maths NCERT Class 10 Chapter 2 – Polynomials Exercise 2.3
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) = x3-3x2+5x–3 , g(x) = x2–2
Solution:
Given,
Dividend = p(x) = x3-3x2+5x–3
Divisor = g(x) = x2– 2
Therefore, upon division, we get
Quotient = x–3
Remainder = 7x–9
(ii) p(x) = x4-3x2+4x+5 , g(x) = x2+1-x
Solution:
Given,
Dividend = p(x) = x4 – 3x2 + 4x +5
Divisor = g(x) = x2 +1-x
Therefore, upon division, we get
Quotient = x2 + x–3
Remainder = 8
(iii) p(x) =x4–5x+6, g(x) = 2–x2
Solution:
Given,
Dividend = p(x) =x4 – 5x + 6 = x4 +0x2–5x+6
Divisor = g(x) = 2–x2 = –x2+2
Therefore, upon division, we get
Quotient = -x2-2
Remainder = -5x + 10
2. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.
(i) t2-3, 2t4 +3t3-2t2-9t-12
Solutions:
Given,
First polynomial = t2-3
Second polynomial = 2t4 +3t3-2t2 -9t-12
As we can see, the remainder is left as 0. Therefore, we say that t2-3 is a factor of 2t4 +3t3-2t2 -9t-12
(ii)x2+3x+1 , 3x4+5x3-7x2+2x+2
Solutions:
Given,
First polynomial = x2+3x+1
Second polynomial = 3x4+5x3-7x2+2x+2
As we can see, the remainder is left as 0. Therefore, we say that x2 + 3x + 1 is a factor of 3x4+5x3-7x2+2x+2
(iii) x3-3x+1, x5-4x3+x2+3x+1
Solutions:
Given,
First polynomial = x3-3x+1
Second polynomial = x5-4x3+x2+3x+1
As we can see, the remainder is not equal to 0. Therefore, we say that x3-3x+1 is not a factor of x5-4x3+x2+3x+1
3. Obtain all other zeroes of 3x4+6x3-2x2-10x-5, if two of its zeroes are √(5/3) and – √(5/3).
Solutions:
Since this is a polynomial equation of degree 4, there will be a total of 4 roots.
√(5/3) and – √(5/3) are zeroes of polynomial f(x).
∴ (x –√(5/3)) (x+√(5/3) = x2-(5/3) = 0
(3x2−5)=0, is a factor of given polynomial f(x).
Now, when we will divide f(x) by (3x2−5), the quotient obtained will also be a factor of f(x), and the remainder will be 0.
Therefore, 3x4 +6x3 −2x2 −10x–5 = (3x2 –5)(x2+2x+1)
Now, on further factorising (x2+2x+1), we get
x2+2x+1 = x2+x+x+1 = 0
x(x+1)+1(x+1) = 0
(x+1)(x+1) = 0
So, its zeroes are given by: x= −1 and x = −1
Therefore, all four zeroes of the given polynomial equation are
√(5/3),- √(5/3) , −1 and −1
Hence, the above-given solution is the answer.
4. On dividing x3-3x2+x+2 by a polynomial g(x), the quotient and remainder were x–2 and –2x+4, respectively. Find g(x).
Solution:
Given,
Dividend, p(x) = x3-3x2+x+2
Quotient = x-2
Remainder = –2x+4
We have to find the value of Divisor, g(x) =?
As we know,
Dividend = Divisor × Quotient + Remainder
∴ x3-3x2+x+2 = g(x)×(x-2) + (-2x+4)
x3-3x2+x+2-(-2x+4) = g(x)×(x-2)
Therefore, g(x) × (x-2) = x3-3x2+3x-2
Now, for finding g(x), we will divide x3-3x2+3x-2 with (x-2)
Therefore, g(x) = (x2–x+1)
5. 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)
(ii) deg q(x) = deg r(x)
(iii) deg r(x) = 0
Solutions:
According to the division algorithm, dividend p(x) and divisor g(x) are two polynomials, where g(x)≠0. Then, we can find the value of quotient q(x) and remainder r(x) with the help of below-given formula.
Dividend = Divisor × Quotient + Remainder
∴ p(x) = g(x)×q(x)+r(x)
Where r(x) = 0 or degree of r(x)< degree of g(x)
Now, let us prove the three given cases, as per the division algorithm, by taking examples for each.
(i) deg p(x) = deg q(x)
The degree of dividend is equal to the degree of the quotient only when the divisor is a constant term.
Let us take an example: p(x) = 3x2+3x+3 is a polynomial to be divided by g(x) = 3
So, (3x2+3x+3)/3 = x2+x+1 = q(x)
Thus, you can see the degree of quotient q(x) = 2, which is also equal to the degree of dividend p(x).
Hence, the division algorithm is satisfied here.
(ii) deg q(x) = deg r(x)
Let us take an example: p(x) = x2 + 3 is a polynomial to be divided by g(x) = x – 1
So, x2 + 3 = (x – 1)×(x) + (x + 3)
Hence, quotient q(x) = x
Also, remainder r(x) = x + 3
Thus, you can see the degree of quotient q(x) = 1, which is also equal to the degree of remainder r(x).
Hence, the division algorithm is satisfied here.
(iii) deg r(x) = 0
The degree of remainder is 0 only when the remainder left after the division algorithm is constant.
Let us take an example: p(x) = x2 + 1 is a polynomial to be divided by g(x) = x.
So, x2 + 1 = (x)×(x) + 1
Hence, quotient q(x) = x
And, remainder r(x) = 1
Clearly, the degree of remainder here is 0.
Hence, the division algorithm is satisfied here.
NCERT Solutions for Class 10 Maths Chapter 2 – Polynomials Ex 2.3 covers the following topics.
- Division Algorithm for Polynomials – It includes five questions in which the first, second and fifth ones have three subparts each.
Key Features of NCERT Solutions for Class 10 Maths Chapter 2 – Polynomials Exercise 2.3
- These NCERT Class 10 Maths Solutions help students solve and revise all questions of this particular exercise 2.3 and, thereby, understand the concept better.
- Students will be able to score good marks after going through the step-wise solutions given by our subject expert teachers.
- They help in scoring well in the Maths board exams.
- They follow NCERT guidelines, which help in preparing the students accordingly.
- They contain all the important questions from the examination point of view.
Very good
Great help!
Great help good explanation 👍🏻