if alpha and beta are zeros of the polynomial f(x)=xx+px+q then polynomial having 1/alpha and 1/beta as its zeros is:
To find the polynomial with 1/alpha and 1/beta as its zeros, we can make use of the concept of Vieta's formulas.
Vieta's formulas state that for a polynomial of the form f(x) = ax^2 + bx + c, if alpha and beta are the zeros, then:
alpha + beta = -b/a (Sum of zeros)
alpha * beta = c/a (Product of zeros)
Using this information, let's proceed to derive the polynomial.
Given:
Zeros of f(x) are alpha and beta.
We know that alpha * beta = q/1 (Product of zeros)
To find the polynomial with zeros 1/alpha and 1/beta, we need to determine the coefficients of the polynomial. Let's suppose this polynomial is g(x).
The zeros of g(x) are 1/alpha and 1/beta, which imply that:
1/alpha + 1/beta = -coefficient of x / constant term (Sum of zeros)
Now, let's substitute the values:
1/alpha + 1/beta = -(-p) / q
1/alpha + 1/beta = p / q
So, the sum of zeros of g(x) is p/q.
Now, we need to find the product of the zeros of g(x). Since the zeros are 1/alpha and 1/beta:
(1/alpha) * (1/beta) = constant term / coefficient of x
1/(alpha * beta) = constant term / coefficient of x
Simplifying this:
1/(alpha * beta) = 1 / (q/1)
1/(alpha * beta) = 1/q
Therefore, the product of zeros of g(x) is 1/q.
Now we have the sum of zeros (p/q) and the product of zeros (1/q). We can form a quadratic polynomial using these values.
The quadratic polynomial g(x) will have zeros 1/alpha and 1/beta and can be written as:
g(x) = x^2 - (p/q)x + (1/q)
So, the polynomial with 1/alpha and 1/beta as its zeros is g(x) = x^2 - (p/q)x + (1/q).
suppose it 'a' as alpha and 'b' as beta
a+b=-(-p/1)=p
and
ab=q/1=q
1/a+1/b=a+b/ab=p/q
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