Certainly! Let's solve the problem step-by-step and construct a quadratic polynomial given its roots [tex]\(3 + \sqrt{2}\)[/tex] and [tex]\(3 - \sqrt{2}\)[/tex].
### Step 1: Understand the Roots
We are given two roots for the quadratic polynomial:
- [tex]\( \text{root}_1 = 3 + \sqrt{2} \)[/tex]
- [tex]\( \text{root}_2 = 3 - \sqrt{2} \)[/tex]
### Step 2: Sum and Product of the Roots
For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] with roots [tex]\( r_1 \)[/tex] and [tex]\( r_2 \)[/tex], the sum and product of the roots are given by the following relations:
- Sum of roots [tex]\( (r_1 + r_2) \)[/tex]
- Product of roots [tex]\( (r_1 \times r_2) \)[/tex]
Calculate the sum of the roots:
[tex]\[ r_1 + r_2 = (3 + \sqrt{2}) + (3 - \sqrt{2}) = 6 \][/tex]
Calculate the product of the roots:
[tex]\[ r_1 \times r_2 = (3 + \sqrt{2})(3 - \sqrt{2}) \][/tex]
Using the difference of squares formula:
[tex]\[ r_1 \times r_2 = 3^2 - (\sqrt{2})^2 = 9 - 2 = 7 \][/tex]
### Step 3: Form the Quadratic Polynomial
Using the standard form of a quadratic polynomial:
[tex]\[ x^2 - ( \text{sum of roots} ) x + ( \text{product of roots} ) \][/tex]
Substitute the sum and product of the roots:
[tex]\[ x^2 - 6x + 7 \][/tex]
### Conclusion:
The quadratic polynomial with the roots [tex]\( 3 + \sqrt{2} \)[/tex] and [tex]\( 3 - \sqrt{2} \)[/tex] is:
[tex]\[ \boxed{x^2 - 6x + 7} \][/tex]
This polynomial has a sum of roots equal to [tex]\(6\)[/tex] and a product of roots equal to [tex]\(7\)[/tex].
