(i) 1/4, -1 (ii) √2, 1/3 (ii) 0, √5 (iv) 1, 1 (v) -1/4, 1/4 (vi) 4, 1
Solution:
(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 – (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 – (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 – (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 – (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–(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 = 4
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.