NCERT Solutions for Chapter 2 Polynomials Class 10 Maths

NCERT Solution for Class 10 Mathematics Chapter 2 Polynomials

Chapter Name	NCERT Solution for Class 10 Maths Chapter 2 Polynomials
Topics Covered	Short Revision for the Chapter NCERT Exercise Solution
Related Study	NCERT Solution for Class 10 Maths NCERT Revision Notes for Class 10 Maths Important Questions for Class 10 Maths MCQ for Class 10 Maths NCERT Exemplar Questions For Class 10 Maths

Short Revision for Polynomials

1. In a polynomial p(x), the highest exponent of x is called the degree of the polynomial.
2. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.
3. 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 Sled the value of p(x) at x = k, and is denoted by p(k).
4. If on substituting x = k in a polynomial p(x), we get p(k) = 0, then k is said to be a zero of the polynomial.
5. Every real number is a constant polynomial.
6. 0 is the zero polynomial.
7. The degree of a non-zero constant polynomial is zero.
8. Polynomials of one term, two terms and three terms are called monomial, binomial and trinomial respectively. 
9. 2x³ + 5x² – 7x + √3 is a polynomial in the variable x of degree 3. 
10. x^5/2 + x² – 7x + 3 is not a polynomial. 
11. If the graph of a polynomial intersects x - axis at n points, then the number of zeroes of the polynomial is n. 
12. If a linear polynomial is p(x) = ax + b, then zero of the polynomial = -(Constant term)/Coefficient of x  =  -b/a.
13. If a quadratic polynomial is p(x) = ax² + bx + c, then
    Sum of zeroes = -(coefficient of x)/coefficient of x²) = -b/a
    Product of zeroes = Constant term/Coefficient of x² = c/a.
14. If a cubic polynomial is p(x) = ax³ + bx² + cx + d, then
    Sum of zeroes = -(Coefficient of x²)/(Coefficient of x³) = -b/a
    Sum of the product of zeroes taken two at a time = (Coefficient of x)/(Coefficient of x²) = c/a
    Product of zeroes = -(Constant term)/(Coefficient of x²) = -d/a
15. If one polynomial P(x)is divided by the other polynomial g(x)≠0, then the relation among p(x), g(x), quotient q(x) and remainder r(x) is given by
    p(x) = g(x) × q(x) + r(x), where degree of r(x) < degree of g(x).
    i.e., Dividend  = Divisor × Quotient + Remainder 
16. A linear polynomial has at most 1 zero. 
17. A quadratic polynomial has at most 2 zeroes. 
18. A cubic polynomial has at most 3 zeroes.

NCERT Exercises Solution

Exercise 2.1

1. The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

Solution

(i) As the graph of polynomial does not meet x - axis, so the polynomial has no zeroes. 
(ii) As the graph of polynomial cuts (meets) x - axis only once, so the polynomial has exactly one zero. 
(iii) As the graph of polynomial cuts (meets) x - axis thrice, so the polynomial has three zeroes. 
(iv) As the graph of polynomial cuts(meets) x - axis twice, so the polynomial has exactly two zeroes. 
(v) As the graph of polynomial cuts (meets) x - axis four times, so the polynomial has four zeroes.
(vi) As the graph of polynomial cuts (meets) x- axis three times, so the polynomial has three zeroes. 

Exercise 2.2

1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) x²–2x –8
(ii) 4s²–4s+1
(iii) 6x²–3–7x
(iv) 4u²+8u
(v) t²–15
(vi) 3x²–x–4

2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i) 1/4 , -1
(ii)√2, 1/3
(iii) 0, √5
(iv) 1, 1
(v) -1/4, 1/4
(vi) 4, 1

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
(ii) p(x) = x⁴-3x²+4x+5 , g(x) = x²+1-x
(iii) p(x) =x⁴–5x+6, g(x) = 2–x²

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
(ii)x²+3x+1 , 3x⁴+5x³-7x²+2x+2
(iii) x³-3x+1, 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).

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).

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

Exercise 2.4 

1. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:
(i) 2x³+x²-5x+2; -1/2, 1, -2
(ii) x³-4x²+5x-2 ;2, 1, 1

2. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively.

3. If the zeroes of the polynomial x³-3x²+x+1 are a – b, a, a + b, find a and b.

4. If two zeroes of the polynomial x⁴-6x³-26x²+138x-35 are 2 ±√3, find other zeroes.

5. If the polynomial x⁴ + - 6x³ + 16x² – 25x + 10 is divided by another polynomial x² – 2x + k, the remainder comes out to be x + a, find k and a.