if alpha and beta are the zeros of polynomial x^2-4x+3 form a quadratic polynomial whose zeros are 1/α and 1/β
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if alpha and beta are the zeros of polynomial x^2-4x+3 form a quadratic polynomial whose zeros are 1/α and 1/β
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If α and β are the zeros of the polynomial x^2 - 4x + 3, we can find a quadratic polynomial whose zeros are 1/α and 1/β by reversing the roles of α and β.
The original polynomial is:
x^2 - 4x + 3 = 0
Now, let's switch α and β:
(x - α)(x - β) = 0
Now, we want to find a quadratic polynomial with roots 1/α and 1/β. To do this, we'll find the reciprocals of α and β:
1/α and 1/β
To form a new polynomial with these roots, we need to reverse the roles of α and β again:
(x - 1/α)(x - 1/β) = 0
Now, let's expand this expression to get the quadratic polynomial:
(x - 1/α)(x - 1/β) = 0
x^2 - (1/α + 1/β)x + (1/α)(1/β) = 0
So, the quadratic polynomial with roots 1/α and 1/β is:
x^2 - [(1/α) + (1/β)]x + (1/α)(1/β) = 0
You can simplify further if you know the values of α and β.
Explanation:
alpha and beta are the zeroes of polynomial x2-4x+3
we know that ,
sum of zeroes =alpha +beta
=-coeffocient of x/ coefficient of x2
=-b/a
=-(-)4/1
=4
also product of roots =appha ×beta =
coefficient of constant term / coefficient of x2
=3/1
=3
now ,
(alpha -beta)=alpha+beta -4alpha × beta
alpha -beta =4-4×3
=4-12
=-8
now ,
alpha -beta =-8......(1)
alpha +beta =4....(2)
the adding equ.(1) and (2) we get
2alpha=12
alpha =6
putting this value in equation (1)
6-beta=-8
6+8= beta
14=beta
hence,
1/ alpha =1/6
and 1/beta =1/14