NCERT Solutions: Polynomials (Exercise 2.2)

# Polynomials (Exercise 2.2) NCERT Solutions - Mathematics (Maths) Class 10

Q1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

Solutions:
(i) x2–2x –8
⇒ x2– 4x+2x–8 = x(x–4)+2(x–4) = (x-4)(x+2)
Therefore, zeroes of polynomial equation x2–2x–8 are (4, -2)
Sum of zeroes = 4–2 = 2 = -(-2)/1 = -(Coefficient of x)/(Coefficient of x2)
Product of zeroes = 4×(-2) = -8 =-(8)/1 = (Constant term)/(Coefficient of x2)

(ii) 4s2–4s+1
⇒ 4s2–2s–2s+1 = 2s(2s–1)–1(2s-1) = (2s–1)(2s–1)
Therefore, zeroes of polynomial equation 4s2–4s+1 are (1/2, 1/2)
Sum of zeroes = (½)+(1/2) = 1 = -4/4 = -(Coefficient of s)/(Coefficient of s2)
Product of zeros = (1/2)×(1/2) = 1/4 = (Constant term)/(Coefficient of s)

(iii) 6x2–3–7x
⇒ 6x2–7x–3 = 6x– 9x + 2x – 3 = 3x(2x – 3) +1(2x – 3) = (3x+1)(2x-3)
Therefore, zeroes of polynomial equation 6x2–3–7x are (-1/3, 3/2)
Sum of zeroes = -(1/3)+(3/2) = (7/6) = -(Coefficient of x)/(Coefficient of x2)
Product of zeroes = -(1/3)×(3/2) = -(3/6) = (Constant term) /(Coefficient of x)

(iv) 4u2+8u
⇒ 4u(u+2)
Therefore, zeroes of polynomial equation 4u2 + 8u are (0, -2).
Sum of zeroes = 0+(-2) = -2 = -(8/4) = = -(Coefficient of u)/(Coefficient of u2)
Product of zeroes = 0×-2 = 0 = 0/4 = (Constant term)/(Coefficient of u)

(v) t2–15
⇒ t2 = 15 or t = ±√15
Therefore, zeroes of polynomial equation t2 –15 are (√15, -√15)
Sum of zeroes =√15+(-√15) = 0= -(0/1)= -(Coefficient of t) / (Coefficient of t2)
Product of zeroes = √15×(-√15) = -15 = -15/1 = (Constant term) / (Coefficient of t)

(vi) 3x2–x–4
⇒ 3x2–4x+3x–4 = x(3x-4)+1(3x-4) = (3x – 4)(x + 1)
Therefore, zeroes of polynomial equation3x2 – x – 4 are (4/3, -1)
Sum of zeroes = (4/3)+(-1) = (1/3)= -(-1/3) = -(Coefficient of x) / (Coefficient of x2)
Product of zeroes=(4/3)×(-1) = (-4/3) = (Constant term) /(Coefficient of x)

Q2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i) 1/4 , -1
Solution:
From the formulas of sum and product of zeroes, we know,
Sum of zeroes = α+β
Product of zeroes = α β
Sum of zeroes = α+β = 1/4
Product of zeroes = α β = -1
∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:
x2–(α+β)x +αβ = 0
x2–(1/4)x +(-1) = 0
4x2–x-4 = 0

(ii)√2, 1/3
Solution:
Sum of zeroes = α + β =√2
Product of zeroes = α β = 1/3
∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:
x2–(α+β)x +αβ = 0
x2 –(√2)x + (1/3) = 0
3x2-3√2x+1 = 0
Thus, 3x2-3√2x+1 is the quadratic polynomial.

(iii) 0, √5
Solution:
Given,
Sum of zeroes = α+β = 0
Product of zeroes = α β = √5
∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:
x2–(α+β)x +αβ = 0
x2–(0)x +√5= 0
Thus, x2+√5 is the quadratic polynomial.

(iv) 1, 1
Solution:
Given,
Sum of zeroes = α+β = 1
Product of zeroes = α β = 1
∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:
x2–(α+β)x +αβ = 0
x2–x+1 = 0
Thus , x2–x+1is the quadratic polynomial.

(v) -1/4, 1/4
Solution:
Given,
Sum of zeroes = α+β = -1/4
Product of zeroes = α β = 1/4
∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:
x2–(α+β)x +αβ = 0
x2–(-1/4)x +(1/4) = 0
4x2+x+1 = 0

(vi) 4, 1
Solution:
Given,
Sum of zeroes = α+β =
Product of zeroes = αβ = 1
∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:
x2–(α+β)x+αβ = 0
x2–4x+1 = 0
Thus, x2–4x+1 is the quadratic polynomial.

Check out the NCERT Solutions of all the exercises of Polynomials:

Exercise 2.1. NCERT Solutions: Polynomials

Exercise 2.3 NCERT Solutions: Polynomials

Exercise 2.4 NCERT Solutions: Polynomials

The document Polynomials (Exercise 2.2) NCERT Solutions | Mathematics (Maths) Class 10 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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## FAQs on Polynomials (Exercise 2.2) NCERT Solutions - Mathematics (Maths) Class 10

 1. What are polynomials? Ans. Polynomials are algebraic expressions that consist of variables, coefficients, and exponents. They are made up of terms that are combined using addition, subtraction, multiplication, and division. The degree of a polynomial is determined by the highest exponent of the variable. For example, in the polynomial 3x^2 + 5x - 2, the degree is 2.
 2. How do you classify polynomials based on the number of terms? Ans. Polynomials can be classified based on the number of terms they have. A polynomial with only one term is called a monomial. A polynomial with two terms is called a binomial, and a polynomial with three terms is called a trinomial. Polynomials with more than three terms are simply called polynomials.
 3. What is the zero of a polynomial? Ans. The zero of a polynomial is a value of the variable that makes the polynomial equal to zero. In other words, it is the value of the variable that satisfies the equation P(x) = 0, where P(x) is the polynomial. Finding the zeros of a polynomial is a crucial step in solving polynomial equations and understanding the behavior of the polynomial.
 4. How do you add and subtract polynomials? Ans. To add or subtract polynomials, you need to combine like terms. Like terms are the terms that have the same variables raised to the same powers. Add or subtract the coefficients of the like terms while keeping the variables and their exponents the same. If a term appears in one polynomial but not in the other, simply copy it to the resulting polynomial without any changes.
 5. How do you multiply polynomials? Ans. To multiply polynomials, you can use the distributive property or the method of long multiplication. If you are using the distributive property, multiply each term of one polynomial by each term of the other polynomial, and then combine like terms. If you are using the method of long multiplication, multiply each term of one polynomial by each term of the other polynomial and write the results in a grid. Then, add up the terms diagonally to get the final polynomial.

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