If α and β are the Zeros of the Polynomial f(x)=x^2+x−2, Find the Value of (1/α−1/β)

If α and β are the Zeros of the Polynomial f(x)= x²+x−2, Find the Value of (1/α−1/β)

Solution

To find the value of (1/α - 1/β) where α and β are the zeros of the polynomial f(x) = x^2 + x - 2, we need to determine the values of α and β first.

For a quadratic polynomial of the form f(x) = ax² + bx + c, the sum of the zeros is equal to -b/a, and the product of the zeros is equal to c/a.

In this case, we have the quadratic polynomial f(x) = x² + x - 2.

Using the formulas mentioned above: Sum of the zeros = -b/a = -1/1 = -1 Product of the zeros = c/a = -2/1 = -2

Now, let's find the values of α and β.

We know that the sum of the zeros α + β = -1, and the product of the zeros α * β = -2.

We can solve these equations simultaneously to find the values of α and β.

Using the method of factorization or solving the equations, we find that α = 1 and β = -2 or α = -2 and β = 1

The values of α and β can be interchanged, resulting in two possible sets of solutions:

• For α = 1 and β = -2: (1/α - 1/β) = (1/1 - 1/(-2)) = (1 + 1/2) = 3/2.
• For α = -2 and β = 1: (1/α - 1/β) = (1/(-2) - 1/1) = (-1/2 - 1) = -3/2.

Therefore, the value of (1/α - 1/β) can be either 3/2 or -3/2, depending on the order of α and β chosen.

Answer

For α = 1 and β = -2: (1/α - 1/β) = (1/1 - 1/(-2)) = (1 + 1/2) = 3/2.

For α = -2 and β = 1: (1/α - 1/β) = (1/(-2) - 1/1) = (-1/2 - 1) = -3/2.

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