If [tex]\(\alpha, \beta, \gamma\)[/tex] are zeroes of [tex]\(f(x) = x^3 - 5x^2 + 7x - 15\)[/tex], find the value of [tex]\((\alpha \beta)^{-1} + (\beta \gamma)^{-1} + (\gamma \alpha)^{-1}\)[/tex].



Answer :

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]