Let's start by identifying the polynomial and its relationship to its roots. Given the polynomial [tex]\( -3x^2 + x - 5 \)[/tex], we know the roots are [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex].
According to Vieta's formulas for a quadratic polynomial [tex]\(ax^2 + bx + c\)[/tex], the sum and product of its roots can be given by:
1. Sum of the roots [tex]\(\alpha + \beta = -\frac{b}{a}\)[/tex]
2. Product of the roots [tex]\(\alpha\beta = \frac{c}{a}\)[/tex]
In this case, the coefficients of the polynomial [tex]\( -3x^2 + x - 5 \)[/tex] are:
- [tex]\(a = -3\)[/tex]
- [tex]\(b = 1\)[/tex]
- [tex]\(c = -5\)[/tex]
Using the formulas:
1. [tex]\(\alpha + \beta = -\frac{b}{a} = -\frac{1}{-3} = \frac{1}{3}\)[/tex]
2. [tex]\(\alpha \beta = \frac{c}{a} = \frac{-5}{-3} = \frac{5}{3}\)[/tex]
Next, we need to find the value of [tex]\(\frac{1}{\alpha} + \frac{1}{\beta}\)[/tex]. This can be expressed as:
[tex]\[
\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta}
\][/tex]
Substituting the known values:
[tex]\[
\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\frac{1}{3}}{\frac{5}{3}} = \frac{1}{3} \times \frac{3}{5} = \frac{1}{5} = 0.2
\][/tex]
Given the calculations:
- The sum of the roots is [tex]\( \frac{1}{3} \)[/tex].
- The product of the roots is [tex]\( \frac{5}{3} \)[/tex].
- The value of [tex]\( \frac{1}{\alpha} + \frac{1}{\beta} \)[/tex] is [tex]\( 0.2 \)[/tex].
Thus, the final answer is [tex]\(\boxed{0.2}\)[/tex].
