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

By BYJU'S Exam Prep

Updated on: October 17th, 2023

If α and β are the zeros of the polynomial f(x)=x²+x−2, find the value of (1/α−1/β).

Here the steps which will be used to find the desried result:

Step 1: First set the polynomial equal to zero: x² + x – 2 = 0.

Step 2: To find the answer, we will first factorize the quadratic equation or solve it using the quadratic formula.

Step 3: Then substitute the values of α and β into the expression (1/α – 1/β)

Step 4: The value of (1/α−1/β) will be thus obtained.

Table of content

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:

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.

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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