Sure, let's solve this step-by-step.
First, we are given a quadratic polynomial [tex]\(3x^2 + 2x - 6\)[/tex]. We need to find the zeroes [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] of this polynomial and then determine the following expressions involving [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex]:
(a) [tex]\(\alpha + \beta\)[/tex]
(b) [tex]\(\frac{1}{\alpha} + \frac{1}{\beta}\)[/tex]
(c) [tex]\(\alpha + \beta + \alpha \beta\)[/tex]
### Step 1: Find the sum of the roots [tex]\(\alpha + \beta\)[/tex]
For a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex], the sum of the roots ([tex]\(\alpha + \beta\)[/tex]) is given by the formula:
[tex]\[
\alpha + \beta = -\frac{b}{a}
\][/tex]
Here, [tex]\(a = 3\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(c = -6\)[/tex]. Therefore:
[tex]\[
\alpha + \beta = -\frac{2}{3}
\][/tex]
So, [tex]\(\alpha + \beta = -0.6666666666666666\)[/tex].
### Step 2: Find [tex]\(\frac{1}{\alpha} + \frac{1}{\beta}\)[/tex]
To find [tex]\(\frac{1}{\alpha} + \frac{1}{\beta}\)[/tex], we use the identity:
[tex]\[
\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta}
\][/tex]
We already have [tex]\(\alpha + \beta = -\frac{2}{3}\)[/tex].
Next, we need to find [tex]\(\alpha \beta\)[/tex], which for a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
\alpha \beta = \frac{c}{a}
\][/tex]
Here, [tex]\(c = -6\)[/tex] and [tex]\(a = 3\)[/tex]. Therefore:
[tex]\[
\alpha \beta = \frac{-6}{3} = -2
\][/tex]
Now we can compute:
[tex]\[
\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-\frac{2}{3}}{-2} = \frac{-0.6666666666666666}{-2} = 0.3333333333333333
\][/tex]
### Step 3: Find [tex]\(\alpha + \beta + \alpha \beta\)[/tex]
We already have the values [tex]\(\alpha + \beta = -0.6666666666666666\)[/tex] and [tex]\(\alpha \beta = -2\)[/tex]. So, we add them:
[tex]\[
\alpha + \beta + \alpha \beta = -0.6666666666666666 + (-2) = -2.6666666666666665
\][/tex]
### Summary
(a) [tex]\(\alpha + \beta = -0.6666666666666666\)[/tex]
(b) [tex]\(\frac{1}{\alpha} + \frac{1}{\beta} = 0.3333333333333333\)[/tex]
(c) [tex]\(\alpha + \beta + \alpha \beta = -2.6666666666666665\)[/tex]
These are the requested values.
