If 
are the zeros of the quadratic polynomial 
,find the quadratic polynomial whose zeros are 
Answers
SOLUTION :
Given : α and β are the zeroes of the quadratic polynomial f(x) = x² - 1
On comparing with ax² + bx + c,
a = 1 , b = 0 , c = -1
Sum of the zeroes = −coefficient of x / coefficient of x²
α + β = -b/a = 0/1 = 0
α+β = 0 ……………………….(1)
Product of the zeroes = constant term/ Coefficient of x²
αβ = c/a = -1/1 = -1
α×β = - 1 ……………… (2)
A.T.Q
Sum of the zeroes of the required polynomial = 2α/β and 2β/α
= 2α² + 2β² / αβ [ By taking L.C.M]
=2(α² + β² ) / αβ
= 2[(α+β)² –2αβ]/αβ
[ a² + b² = (a + b)² - 2ab ]
= 2[0 - 2 × - 1] /- 1
[From eq 1 & 2]
= 2[ 0 + 2] /-1
= 2 × 2 /- 1
= 4/-1 = - 4
2α/β + 2β/α = - 4 ……………………(3)
Product of the zeroes =2α/β × 2β/α
= 4αβ / αβ = 4
2α/β × 2β/α = 4 ………………….(4)
Hence, the required quadratic polynomial is
k(x²–(sum of the zeroes)x + (product of the zeroes) )
= k(x² - (- 4)x + 4 )
[ From eq 3 & 4 ]
= k(x² + 4x + 4 )
[K is any non zero real number]
Hence, the required quadratic polynomial is f(x)= k(x² + 4x + 4 )
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Answer:
SOLUTION :
Given : α and β are the zeroes of the quadratic polynomial f(x) = x² - 1
On comparing with ax² + bx + c,
a = 1 , b = 0 , c = -1
Sum of the zeroes = −coefficient of x / coefficient of x²
α + β = -b/a = 0/1 = 0
α+β = 0 ……………………….(1)
Product of the zeroes = constant term/ Coefficient of x²
αβ = c/a = -1/1 = -1
α×β = - 1 ……………… (2)
A.T.Q
Sum of the zeroes of the required polynomial = 2α/β and 2β/α
= 2α² + 2β² / αβ [ By taking L.C.M]
=2(α² + β² ) / αβ
= 2[(α+β)² –2αβ]/αβ
[ a² + b² = (a + b)² - 2ab ]
= 2[0 - 2 × - 1] /- 1
[From eq 1 & 2]
= 2[ 0 + 2] /-1
= 2 × 2 /- 1
= 4/-1 = - 4
2α/β + 2β/α = - 4 ……………………(3)
Product of the zeroes =2α/β × 2β/α
= 4αβ / αβ = 4
2α/β × 2β/α = 4 ………………….(4)
Hence, the required quadratic polynomial is
k(x²–(sum of the zeroes)x + (product of the zeroes)
)
= k(x² - (- 4)x + 4
)
[ From eq 3 & 4 ]
= k(x² + 4x + 4
)
[K is any non zero real number]
Hence, the required quadratic polynomial is f(x)= k(x² + 4x + 4
)