In algebra, a quadratic polynomial is a polynomial of degree two. It is of the form ax² + bx + c, where a, b, and c are constants. The zeros of a quadratic polynomial are the values of x that make the polynomial equal to zero. In this problem, we are given one zero of the quadratic polynomial and the sum of the zeroes. We need to use this information to form the quadratic polynomial.



We are given:

- One zero of the quadratic polynomial, which is root2 + 5.
- The sum of the zeroes, which is 4.



Since we are given one zero of the quadratic polynomial, we know that (x - (root2 + 5)) is a factor of the polynomial. To find the other factor, we can use the sum of the zeroes.

The sum of the zeroes of a quadratic polynomial is given by -b/a, where b and a are the coefficients of x and x² respectively. In this case, the sum of the zeroes is 4, so we have:

- (-b/a) = 4

We also know that the coefficient of x² is 1 (since it is not given). Therefore, we can write the quadratic polynomial as:

- (x - (root2 + 5))(x - p) = x² - (p + root2 + 5)x + (root2 + 5)p

where p is the other zero of the polynomial.

To find the value of p, we can substitute the sum of the zeroes in the above equation and solve for p.

- (-b/a) = 4
- => (p + root2 + 5)/1 = -4
- => p + root2 + 5 = -4
- => p = -4 - root2 - 5
- => p = -root2 - 9

Therefore, the quadratic polynomial is:

- (x - (root2 + 5))(x - (-root2 - 9)) = x² - 4x - 11



In this problem, we used the given information of one zero and the sum of the zeroes to form a quadratic polynomial. We found that the quadratic polynomial is x² - 4x - 11.