Answer:
f(x) = x² - 10x + 22
Explanation:
Let's assume the quadratic polynomial as:
f(x) = ax² + bx + c
Now we know that if one of the zeroes is 5 + √3, then the other zero must be 5 - √3 (because complex roots always come in conjugate pairs).
So the sum of the zeroes will be:
(5 + √3) + (5 - √3) = 10
10 = 2 * 5
The product of the zeroes will be:
(5 + √3) * (5 - √3) = 25 - 3 = 22
Now, using the sum and product of zeroes, we can write:
b/a = 10
c/a = 22
Solving for b and c, we get:
b = -10a
c = 22a
Substituting these values in f(x), we get:
f(x) = a(x - 5 - √3)(x - 5 + √3)
Expanding the right-hand side:
f(x) = a[(x - 5)² - (√3)²]
f(x) = a(x² - 10x + 22)
Comparing the coefficients of f(x) with ax² + bx + c, we get:
a = 1, b = -10, c = 22
Therefore, the quadratic polynomial is:
f(x) = x² - 10x + 22