NCERT Solutions for Class 10 Maths Unit 2

Polynomials Class 10

Unit 2 Polynomials Exercise 2.1, 2.2, 2.3 2.4, 2.4 Solutions

In mathematics, a polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single indeterminate (or variable), x, is x2 - 4x + 7, which is a quadratic polynomial.

Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry.

Exercise 2.1 : Solutions of Questions on Page Number : 28

Q1 :

The graphs of y = p(x) are given in following figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

(i) (ii) (iii) (iv) (v) (v) (i) The number of zeroes is 0 as the graph does not cut the x-axis at any point.

(ii) The number of zeroes is 1 as the graph intersects the x-axis at only 1 point.

(iii) The number of zeroes is 3 as the graph intersects the x-axis at 3 points.

(iv) The number of zeroes is 2 as the graph intersects the x-axis at 2 points.

(v) The number of zeroes is 4 as the graph intersects the x-axis at 4 points.

(vi) The number of zeroes is 3 as the graph intersects the x-axis at 3 points.

Exercise 2.2 : Solutions of Questions on Page Number : 33

Q1 :

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.       The value of is zero when x - 4 = 0 or x + 2 = 0, i.e., when x = 4 or x = - 2

Therefore, the zeroes of are 4 and - 2.

Sum of zeroes = Product of zeroes  The value of 4s2 - 4s + 1 is zero when 2s - 1 = 0, i.e., Therefore, the zeroes of 4s2 - 4s + 1 are and .

Sum of zeroes = Product of zeroes  The value of 6x2 - 3 - 7x is zero when 3x + 1 = 0 or 2x - 3 = 0, i.e., or Therefore, the zeroes of 6x2 - 3 - 7x are .

Sum of zeroes = Product of zeroes =  The value of 4u2 + 8u is zero when 4u = 0 or u + 2 = 0, i.e., u = 0 or u = - 2

Therefore, the zeroes of 4u2 + 8u are 0 and - 2.

Sum of zeroes = Product of zeroes =  The value of t2 - 15 is zero when or , i.e., when Q2 :

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.      Exercise 2.3 2.4 : Solutions of Questions on Page Number : 36

Q1 :

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

(i) (ii) (iii) Q2 :

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: Q3 :

Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial: Q4 :

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.

Q5 :

Obtain all other zeroes of , if two of its zeroes are .

Q6 :

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

Q7 :

Give examples of polynomial 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 : Solutions of Questions on Page Number : 37

Q1 :

If the zeroes of polynomial are , find a and b.

Q2 :

]It two zeroes of the polynomial are , find other zeroes.

Q3 :

If the polynomial is divided by another polynomial , the remainder comes out to be x + a, find k and a.