To determine the value of \( k \) for the polynomial \( p(x) = 2x^2 - k\sqrt{2} x + 1 \) given that the sum of its zeroes is \( \sqrt{2} \), let's follow these steps:
1. Recall the general properties of a polynomial:
For any quadratic polynomial \( ax^2 + bx + c \), the sum of the zeroes (roots) is given by the formula:
[tex]\[
\text{Sum of zeroes} = -\frac{b}{a}
\][/tex]
2. Identify the coefficients:
In the polynomial \( p(x) = 2x^2 - k\sqrt{2} x + 1 \), the coefficients are:
[tex]\[
a = 2, \quad b = -k\sqrt{2}, \quad c = 1
\][/tex]
3. Set up the relationship using the sum of the zeroes:
We are given that the sum of the zeroes is \( \sqrt{2} \). According to the formula for the sum of the zeroes, we have:
[tex]\[
-\frac{b}{a} = \sqrt{2}
\][/tex]
4. Substitute the coefficients into the formula:
Substituting \( b = -k\sqrt{2} \) and \( a = 2 \) into the formula, we get:
[tex]\[
-\frac{-k\sqrt{2}}{2} = \sqrt{2}
\][/tex]
5. Simplify the equation:
[tex]\[
\frac{k\sqrt{2}}{2} = \sqrt{2}
\][/tex]
6. Solve for \( k \):
To isolate \( k \), multiply both sides by 2:
[tex]\[
k\sqrt{2} = 2\sqrt{2}
\][/tex]
Now, divide both sides by \( \sqrt{2} \):
[tex]\[
k = \frac{2\sqrt{2}}{\sqrt{2}}
\][/tex]
Simplify the right-hand side:
[tex]\[
k = 2
\][/tex]
Therefore, the value of [tex]\( k \)[/tex] is [tex]\( \boxed{2} \)[/tex].
