Find zeroes of this polynomial and verify the relationship between the zeroes and their coefficients.
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Answers
(1)
The given polynomial is
g(x) = a(x² + 1) - x(a² + 1)
= ax² + a - a²x - x
= ax² - a²x + a - x
= ax (x - a) - 1 (x - a)
= (x - a) (ax - 1)
∴ the zeroes of the given polynomial are a & 1/a
The given polynomial can be written as
g(x) = ax² - (a² + 1)x + a
If α and β be the zeroes of the given polynomial
- α + β = - {- (a² + 1)}/a = (a² + 1)/a
- αβ = a/a = 1
Sum of zeroes = a + 1/a = (a² + 1)/a
Product of zeroes = a * 1/a = 1
Hence, verified.
(2)
The given polynomial is
g(y) = 7y² - (11/3)y - (2/3)
= 7y² - (14/3)y + (3/3)y - (2/3)
= 7y (y - 2/3) + 1 (y - 2/3)
= (y - 2/3) (7y + 1)
∴ the zeroes of the given polynomial are 2/3 & (- 1/7)
If α and β be the zeroes of the given polynomial
- α + β = - {- (11/3)}/7 = 11/21
- αβ = (- 2/3)/7 = - 2/21
Sum of zeroes = 2/3 - 1/7 = 11/21
Product of zeroes = (2/3) (- 1/7) = - 2/21
Hence, verified.