Explanation:

Given quadratic polynomial is x² - x - 2.
Let alpha and beta be the roots of the given quadratic polynomial.

Sum of the roots:
The sum of the roots of the quadratic equation ax² + bx + c = 0 is given by -b/a. Hence, sum of the roots of the given quadratic equation is -(-1)/1 = 1.

Product of the roots:
The product of the roots of the quadratic equation ax² + bx + c = 0 is given by c/a. Hence, product of the roots of the given quadratic equation is -2/1 = -2.

Value of 1/alpha - 1/beta:
We have to find the value of (1/alpha) - (1/beta). Let's simplify this expression.

(1/alpha) - (1/beta) = (beta - alpha)/(alpha*beta)

Using the sum and product of roots, we can write alpha + beta = 1 and alpha * beta = -2.

Hence, (beta - alpha)/(alpha*beta) = (1 - 2alpha)/(alpha*(1-alpha))

Now, we know that alpha and beta are the roots of the given quadratic equation. Hence, we can write the quadratic equation as:

x² - x - 2 = 0

Solving this quadratic equation using the quadratic formula, we get:

alpha = (1 + sqrt(9))/2 = 2
beta = (1 - sqrt(9))/2 = -1

Hence, (1 - 2alpha)/(alpha*(1-alpha)) = (1 - 2*2)/(2*(1-2)) = -3.

Therefore, the value of (1/alpha) - (1/beta) is -3.

X²+x-2
x²+-x-2
x(x+2)-1(x+2)
(x-1)(x+2)
x-1=0 x+2=0
x=1 x=-2
Let α=1 and β=-2
1/α - 1/β
= 1-(1/-2)
=1+(1/2)
=(2+1)/2
=3/2