Certainly! Let's solve the given problem step-by-step.
Given the quadratic polynomial [tex]\(5x^2 + 5x + 1\)[/tex], we need to find the value of [tex]\(\alpha^{-1} + \beta^{-1}\)[/tex] where [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] are the roots of this polynomial.
### Step 1: Understand Vieta's formulas
For a quadratic polynomial of the form [tex]\(ax^2 + bx + c = 0\)[/tex], Vieta's formulas tell us two key relationships between the coefficients of the polynomial and its roots. If [tex]\(\alpha\)[/tex] and \beta are the roots, then:
1. [tex]\( \alpha + \beta = -\frac{b}{a} \)[/tex]
2. [tex]\( \alpha \beta = \frac{c}{a} \)[/tex]
### Step 2: Apply Vieta's formulas to our polynomial
Here, [tex]\(a = 5\)[/tex], [tex]\(b = 5\)[/tex], and [tex]\(c = 1\)[/tex]. Using Vieta's formulas, we can find the sum and product of the roots [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex]:
1. The sum of the roots:
[tex]\[ \alpha + \beta = -\frac{b}{a} = -\frac{5}{5} = -1 \][/tex]
2. The product of the roots:
[tex]\[ \alpha \beta = \frac{c}{a} = \frac{1}{5} \][/tex]
### Step 3: Calculate [tex]\(\alpha^{-1} + \beta^{-1}\)[/tex]
We know:
[tex]\[ \alpha^{-1} + \beta^{-1} \][/tex]
This expression can be rewritten in terms we already have:
[tex]\[ \alpha^{-1} + \beta^{-1} = \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} \][/tex]
Using the values from Vieta's formulas:
- [tex]\(\alpha + \beta = -1\)[/tex]
- [tex]\(\alpha \beta = \frac{1}{5}\)[/tex]
Plugging these values into the equation:
[tex]\[ \alpha^{-1} + \beta^{-1} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-1}{\frac{1}{5}} = -1 \times 5 = -5 \][/tex]
### Final Answer
Hence, the value of [tex]\(\alpha^{-1} + \beta^{-1}\)[/tex] is [tex]\(-5\)[/tex].
