Finding the Roots of the Polynomial
To determine the values of \( \alpha \) and \( \beta \) for the polynomial \( x^2 - 5x + 6 \), we can use the factorization method.
- The polynomial can be rewritten as:
\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\]
- From this factorization, we identify the roots:
- \( \alpha = 2 \)
- \( \beta = 3 \)
Calculating \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \)
Next, we compute \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \):
- The reciprocal of \( \alpha \):
\[
\frac{1}{\alpha} = \frac{1}{2}
\]
- The reciprocal of \( \beta \):
\[
\frac{1}{\beta} = \frac{1}{3}
\]
Finding the Sum of Reciprocals
Now, we can find the sum of the reciprocals:
- The expression becomes:
\[
\frac{1}{\alpha} + \frac{1}{\beta} = \frac{1}{2} + \frac{1}{3}
\]
- Finding a common denominator (which is 6):
\[
\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}
\]
Final Value
Thus, the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) is:
\[
\frac{5}{6}
\]
This approach illustrates how to analyze a quadratic polynomial, find its roots, and compute the sum of the reciprocals of those roots effectively.