To form a quadratic polynomial with zeros α and [tex]\( \frac{1}{\beta} \)[/tex], we need to follow the steps below. We start by finding the roots of the given quadratic equation and then form the new polynomial.
### Step 1: Find the roots (α and β) of the quadratic equation [tex]\(x^2 + \sqrt{2} x + 3 = 0\)[/tex].
The roots of the quadratic equation [tex]\( x^2 + \sqrt{2} x + 3 = 0 \)[/tex] are given by:
[tex]\[ \alpha = -\frac{\sqrt{2}}{2} - \frac{\sqrt{10}i}{2} \][/tex]
[tex]\[ \beta = -\frac{\sqrt{2}}{2} + \frac{\sqrt{10}i}{2} \][/tex]
### Step 2: Form a new quadratic polynomial with roots α and [tex]\( \frac{1}{\beta} \)[/tex].
To form the new quadratic polynomial with these roots, we first need to determine [tex]\( \frac{1}{\beta} \)[/tex]:
[tex]\[ \frac{1}{\beta} = \frac{1}{-\frac{\sqrt{2}}{2} + \frac{\sqrt{10}i}{2}} \][/tex]
### Step 3: Write the new quadratic polynomial.
The general form of a quadratic polynomial with roots [tex]\( p \)[/tex] and [tex]\( q \)[/tex] is:
[tex]\[ (x - p)(x - q) \][/tex]
Substituting [tex]\( p = \alpha \)[/tex] and [tex]\( q = \frac{1}{\beta} \)[/tex]:
[tex]\[ (x - \alpha)\left( x - \frac{1}{\beta} \right) \][/tex]
### Step 4: Express and simplify the new quadratic polynomial.
First, substitute the roots α and [tex]\( \frac{1}{\beta} \)[/tex]:
[tex]\[ \left( x - \left( -\frac{\sqrt{2}}{2} - \frac{\sqrt{10}i}{2} \right) \right) \left( x - \frac{1}{-\frac{\sqrt{2}}{2} + \frac{\sqrt{10}i}{2}} \right) \][/tex]
We can simplify this further by treating it as a general multiplication of binomials and carrying out the multiplication and simplification.
Upon expanding and simplifying the above expression, we get the new quadratic polynomial.
### Step 5: Final expansion and simplification.
After the steps above, the new quadratic polynomial can be simplified to:
[tex]\[ x^2 + \frac{\sqrt{2}}{2}x - x \left( \frac{1}{-\frac{\sqrt{2}}{2} + \frac{\sqrt{10}i}{2}} \right) + \frac{\sqrt{10}i}{2}x - \frac{\sqrt{10}i}{-\frac{\sqrt{2}}{2} + \frac{\sqrt{10}i}{2}} - \frac{\sqrt{2}}{-\frac{\sqrt{2}} + \sqrt{10}i} \][/tex]
### Final Answer
Thus, the new quadratic polynomial with zeros α and [tex]\( \frac{1}{\beta} \)[/tex] is:
[tex]\[ x^2 + \frac{\sqrt{2}}{2}x - x \left( \frac{1}{-\frac{\sqrt{2}}{2} + \frac{\sqrt{10}i}{2}} \right) + \frac{\sqrt{10}i}{2}x - \frac{\sqrt{10}i}{-\frac{\sqrt{2}} + \sqrt{10}i} - \frac{\sqrt{2}}{-\frac{\sqrt{2}} + \sqrt{10}i} \][/tex]
