Chapter 2- Polynomials Exercise 2.2 – NCERT Solutions for class 10 Maths
These solutions are produced by math professionals who have had them evaluated on a regular basis. Students can use these NCERT chapter-by-chapter solutions to study and prepare for their CBSE board exams.
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Question 1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and their coefficients:
(i) x2 – 2x – 8
x2 − 4x + 2x − 8 = 0
x(x−4) + 2(x−4) = 0
(x−4)(x+2) = 0
(x−4) = 0, (x+2) = 0
x = 4, x = −2 are the zeroes of the polynomial
Relationship between the zeroes and the coefficients
Sum of zeroes = −coefficient of x / coefficient of x2
α + β = −b/a
−2+4 = − (−2)/1
2 = 2
Product of zeroes = constant term / coefficient of x2
α.β = c/a
−2×4 = −8/1
−8 = −8
Hence, Verified.
(ii) 4s2 – 4s + 1
4s2−2s−2s+1= 0
2s(2s−1)−(2s−1) = 0
(2s−1)(2s−1) = 0
2s−1= 0, 2s−1 = 0
s =1/2, s = 1/2 are the zeroes of the polynomial.
Relationship between the zeroes and the coefficients
Sum of zeroes =−coefficient of x / coefficient of x2
α + β = −b/a
1/2 + 1/2 = −(−4)/4
1 = 1
Product of zeroes = constant term / coefficient of x2
α.β = c/a
1/2×1/2 = 1/4
1/4 = 1/4
Hence, Verified.
(iii) 6x2 – 3 – 7x
6x2−7x−3 = 0
6x2−9x+2x−3 = 0
3x(2x−3)+(2x−3) = 0
(2x−3) = 0, (3x+1) = 0
x = 3/2, x = −1/3 are the zeroes of the polynomial
Relationship between the zeroes and the coefficients
Sum of zeroes =− coefficient of x / coefficient of x2
α + β = −b/a
α + β = −(−7)/6
3/2 + −1/3 = (7)/6
7/6 = 7/6
Product of zeroes = constant term / coefficient of x2
α.β = c/a
3/2 × −1/3 = (−3)/6
(−3)/6 = (−3)/6
(−1)/2 = (−1)/2
Hence, Verified.
(iv) 4u2 + 8u
4u(u+2) = 0
4u = 0 or u+2 = 0
u = 0 or u = −2 are the zeroes of the polynomial
Relationship between the zeroes and the coefficients
Sum of zeroes =− coefficient of x / coefficient of x2
α + β = −b/a
α + β = −(8)/4
0+(−2) =−2
−2 =−2
Product of zeroes = constant term / coefficient of x2
α.β = c/a
0 ×−2 = 0/4
0 = 0
Hence, Verified.
(v) t2 – 15
t2−15 = 0
t = √15
t = −√15, t = +√15 are the zeroes of the polynomial
Relationship between the zeroes and the coefficients
Sum of zeroes =− coefficient of x / coefficient of x2
α + β =−b/a
α + β = 0/1
−√15+√15 = 0
0 = 0
Product of zeroes = constant term / coefficient of x2
α.β = c/a
−√15 ×√15 =−15/1
−15=−15
Hence, Verified.
(vi) 3x2 – x – 4
3x2−x−4 = 0
3x2−4x+3x−4 = 0
x(3x−4)+(3x−4) = 0
(x+1)(3x−4) = 0
(x+1) = 0 or (3x−4) = 0
x =−1 or x = 4/3 are the zeroes of the polynomial
Relationship between the zeroes and the coefficients
Sum of zeroes =− coefficient of x / coefficient of x2
α + β =−1/3
−1+4/3=−1/3
1/3=1/3
Product of zeroes = constant term / coefficient of x2
α.β = c/a
−1×4/3=−4/3
−4/3=−4/3
Hence, Verified.
Question 2. Find a quadratic polynomial each with the given numbers as the sum and product of zeroes respectively:
![](https://neutronclasses.com/wp-content/uploads/2021/07/Capture2.2-11-1.png)
(i) 1/4 , -1
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−(sum of roots)x + product of roots = 0
x2–(α+β)x +αβ = 0
x2–(1/4)x +(-1) = 0
4x2–x-4 = 0
Thus, 4x2–x–4 is the quadratic polynomial.
(ii)√2, 1/3
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−(sum of roots)x + product of roots = 0
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
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−(sum of roots)x + product of roots = 0
x2–(α+β)x +αβ = 0
x2–(0)x +√5= 0
Thus, x2+√5 is the quadratic polynomial.
(iv) 1, 1
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−(sum of roots)x + product of roots = 0
x2–(α+β)x +αβ = 0
x2–x+1 = 0
Thus , x2–x+1 is the quadratic polynomial.
(v) -1/4, 1/4
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−(sum of roots)x + product of roots = 0
x2–(α+β)x +αβ = 0
x2–(-1/4)x +(1/4) = 0
4x2+x+1 = 0
Thus,4x2+x+1 is the quadratic polynomial.
(vi) 4, 1
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−(sum of roots)x + product of roots = 0
x2–(α+β)x+αβ = 0
x2–4x+1 = 0
Thus, x2–4x+1 is the quadratic polynomial.
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