To solve this problem, we need to analyze the polynomial [tex]\( f(x) = x^3 - 5x^2 + 7x - 15 \)[/tex] and find the value of [tex]\((\alpha \beta)^{-1} + (\beta \gamma)^{-1} + (\gamma \alpha)^{-1}\)[/tex], where [tex]\(\alpha\)[/tex], [tex]\(\beta\)[/tex], and [tex]\(\gamma\)[/tex] are the roots of the polynomial.
We start by using Vieta's formulas, which relate the coefficients of the polynomial to sums and products of its roots. For the polynomial [tex]\( f(x) = x^3 - 5x^2 + 7x - 15 \)[/tex]:
1. The sum of the roots is given by:
[tex]\[
\alpha + \beta + \gamma = 5
\][/tex]
2. The sum of the product of the roots taken two at a time is given by:
[tex]\[
\alpha\beta + \beta\gamma + \gamma\alpha = 7
\][/tex]
3. The product of the roots is:
[tex]\[
\alpha\beta\gamma = 15
\][/tex]
Next, we need to compute the value of [tex]\((\alpha \beta)^{-1} + (\beta \gamma)^{-1} + (\gamma \alpha)^{-1}\)[/tex]. We know that:
[tex]\[
(\alpha \beta)^{-1} + (\beta \gamma)^{-1} + (\gamma \alpha)^{-1} = \frac{1}{\alpha \beta} + \frac{1}{\beta \gamma} + \frac{1}{\gamma \alpha}
\][/tex]
To combine these fractions, recall the property of reciprocals of products. We can express this sum as:
[tex]\[
\frac{\gamma}{\alpha\beta\gamma} + \frac{\alpha}{\beta\gamma\alpha} + \frac{\beta}{\gamma\alpha\beta}
\][/tex]
Simplifying each term:
[tex]\[
\frac{\gamma}{\alpha\beta\gamma} = \frac{1}{\alpha\beta}, \quad \frac{\alpha}{\beta\gamma\alpha} = \frac{1}{\beta\gamma}, \quad \frac{\beta}{\gamma\alpha\beta} = \frac{1}{\gamma\alpha}
\][/tex]
This can be rewritten as:
[tex]\[
\frac{\gamma + \alpha + \beta}{\alpha \beta \gamma}
\][/tex]
Given that [tex]\(\alpha + \beta + \gamma = 5\)[/tex] and [tex]\(\alpha \beta \gamma = 15\)[/tex]:
[tex]\[
\frac{\alpha + \beta + \gamma}{\alpha \beta \gamma} = \frac{5}{15} = \frac{1}{3}
\][/tex]
Thus, the value of [tex]\((\alpha \beta)^{-1} + (\beta \gamma)^{-1} + (\gamma \alpha)^{-1}\)[/tex] is:
[tex]\[
\boxed{\frac{1}{3}}
\][/tex]
