Answer:
The polynomial p(x) can be rewritten as:
p(x) = (x - 4)((3 + sqrt(33))/2 - (3 - sqrt(33))/2)
= (x - 4)(sqrt(33))
The polynomial p(x) can also be rewritten as:
p(x) = (x - 4)(sqrt(11) - sqrt(3))(sqrt(11) + sqrt(3))
= (x - 4)(sqrt(11) - sqrt(3))(sqrt(11) + sqrt(3))
This shows that the polynomial p(x) can be written as the product of linear factors.
Explanation:
To rewrite the polynomial function p(x) as the product of linear factors, we need to factor out the known factor of (x - 4) from the polynomial. The polynomial p(x) can be written as:
p(x) = (x - 4)(x^2 - 3x - 6)
The polynomial x^2 - 3x - 6 can be rewritten as the product of two linear factors using the quadratic formula:
x^2 - 3x - 6 = 0
x = (3 +/- sqrt(3^2 - 4 * 1 * -6)) / (2 * 1)
x = (3 +/- sqrt(9 + 24)) / 2
x = (3 +/- sqrt(33)) / 2