#### 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.**

- Graph of y = p(x) does not intersect the x-axis. Hence, polynomial p(x) has no zero.
- 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.
- Graph of y = p(x) intersects the x-axis at exactly three points. Hence, there are three real zeroes of polynomial p(x).
- Graph of y = p(x) intersect the x-axis at two points and hence two real zeroes of p(x).
- Polynomial p(x) has four real zeroes.
- 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 2Â±A find other zeroes.**

**Question 5. If the polynomial x^{4}**–

**6×3**+

**16×2 –**–

*25x*+ 10 is divided by another polynomial x^{2}**+**

*2x***+**

*k,*the remainder comes out to be*x*

*a,*find*k*and*a.***Related Articles:**

- NCERT Solutions for Class 10
- NCERT Solutions for Class 10 Maths
- Polynomials Class 10 Notes Maths Chapter 2