NCERT Solutions for class 10th Maths Chapter 2 Polynomials

Exercise 2.1

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

Sol.

1. Graph of y = p(x) does not intersect the x-axis. Hence, polynomial p(x) has no zero.
2. Graph of y = p(x) intersects the x-axis at one and only one point. Hence, polynomial p(x) has one and only one real zero.
3. Graph of y = p(x) intersects the x-axis at exactly three points. Hence, there are three real zeroes of polynomial p(x).
4. Graph of y = p(x) intersect the x-axis at two points and hence two real zeroes of p(x).
5. Polynomial p(x) has four real zeroes.
6. Polynomial p(x) has three real zeroes.

Exercise 2.2

Question 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) 6×2-3-7x (iv) 4u2+Su
(v) t2-15 (vi) 3×2-x-4

Exercise 2.3

Question 1. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following :

Note: We stop the division process when either the remainder is zero or its degree is less than the degree of the division.

Note: To carry out division, first write both the dividend and divisor in standard forms.

(iii) To carry out the division, we first write divisor in the standard form. So, divisor=-x2+2
We have,

Question 2. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

Question 4. On dividing x3-3×2+x+2 by a polynomial g(x), the quotient and remainder were x-2and-2x+4, respectively. Find g(x)

Question 5. Give example of polynomials p(x), g(x), q(x), r(x) which satisfy the division algorithm and

(i) degree p(x) = degree q(x)
(ii) degree q(x) = deg r(x)
(iii) degree r(x) = 0

Sol. (i) We can write
2×2-12x+ 10=2(x2-6x+ 5)+0
Here, p(x) = 2×2- 12x + 10 q(x)=x2-6x+5
r(x)=0, g(x)=2

(ii) 2×3-4×2+5=(2×2- 1)(x-2)+(x + 3)
Here, p(x)=2×3-4×2+5
g(x)=2×2-l [degree p(x) = 2]
q(x)=x-2 [degree q(x) = 2]
r(x)=x+3

(iii) x3 -11= (x-l)(x2 +x +1)+(-10)
Here, p(x)=x3-11,
q(x)=x2 +x +1
g(x) = x-1, [degree q(x) = l]
r(x) =-10 [degreer(x) = l]
So, degree r(x) =

Exercise 2.4

Question 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:

Question 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.

Question 3. If the zeroes of the polynomial x3-3×2+x+1 are a-b,a,a+b.Find a and b.

Question 4. If two zeroes of the polynomial x4-6×3-26x+138x-35 are 2±A find other zeroes.

Question 5. If the polynomial x46×3 + 16×2 – 25x + 10 is divided by another polynomial x22x + k, the remainder comes out to be x + a, find k and a.

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