Due to corona pandemic in India many workers migrated to their village .Mr. P.S.Kusuwah from Gwalior decided to help them with food packets and clothes . food packets and clothes donated by them can be represented by the zeros ( alpha beta ) of the polynomial p ( X ) = X2 - X - 3 . Madhav which is student of Kushwah ji also got inspired by him and donated the food packets and clothes in the form of a polynomial whose zeros are 1 + 2 alpha and 1 + 2 beta.
( 1 ) Sum of zeroes of the polynomial whose zeros Madhav donated food packets and clothes ?
( 2 ) Product of zeroes of the polynomial whose zeroes are 1 + 2 alpha and 1 + 2 beta in the form of which Madhav donated food packets and clothes?
( 3 ) Write the actual polynomial of madhav's donation ?
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Answer:
The actual polynomial of Madhav's donation is (X−1−2α)(X−1−2β)
Step-by-step explanation:
Given the polynomial [tex]p(X)=X^2-X-3[/tex] with zeros α and β.
For Madhav's polynomial, the zeros are 1+2α and 1+2β.
Let's address the questions one by one:
(1) Sum of zeroes of Madhav's polynomial:
The sum of the zeros of a polynomial is given by the negation of the coefficient of the linear term divided by the coefficient of the quadratic term.
For Madhav's polynomial:
Sum of zeros = Coefficient of linear term / Coefficient of quadratic term
Sum of zeros = 1+2α+1+2β / 1
Sum of zeros = 2+2(α+β) / 1
Sum of zeros = −2−2(α+β)
(2) Product of zeroes of Madhav's polynomial:
The product of the zeros of a polynomial is given by the constant term divided by the coefficient of the quadratic term.
For Madhav's polynomial:
Product of zeros = Constant term / Coefficient of quadratic term
Product of zeros = (1+2α)(1+2β) / 1
Product of zeros = (1+2α)(1+2β)
(3) Actual polynomial of Madhav's donation:
Since the zeros of the polynomial are 1+2α and 1+2β, the polynomial can be written as: (X−(1+2α))(X−(1+2β))
Expanding the above expression: (X−1−2α)(X−1−2β)
So, the actual polynomial of Madhav's donation is (X−1−2α)(X−1−2β).
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