To determine the other zeros of the polynomial function of degree 4 with rational coefficients, we need to rely on two important properties:
1. Complex Conjugate Root Theorem: If a polynomial has real coefficients and a complex number [tex]\( a + bi \)[/tex] (where [tex]\( a, b \)[/tex] are real numbers and [tex]\( i \)[/tex] is the imaginary unit) as a root, then its complex conjugate [tex]\( a - bi \)[/tex] is also a root.
2. Conjugate Root Theorem for Irrational Numbers: If a polynomial has rational coefficients and an irrational root involving a square root [tex]\( a + \sqrt{b} \)[/tex] (where [tex]\( a, b \)[/tex] are real numbers and [tex]\( \sqrt{b} \)[/tex] is irrational), then its conjugate [tex]\( a - \sqrt{b} \)[/tex] is also a root.
Given the zeros [tex]\( 2 + 5i \)[/tex] and [tex]\( 2 + \sqrt{5} \)[/tex]:
1. Conjugate of [tex]\( 2 + 5i \)[/tex]:
Since [tex]\( 2 + 5i \)[/tex] is a root and the coefficients of the polynomial are real, its complex conjugate [tex]\( 2 - 5i \)[/tex] must also be a root.
2. Conjugate of [tex]\( 2 + \sqrt{5} \)[/tex]:
Similarly, since [tex]\( 2 + \sqrt{5} \)[/tex] is a root and the coefficients of the polynomial are rational, its conjugate [tex]\( 2 - \sqrt{5} \)[/tex] must also be a root.
Thus, the other zeros of the polynomial are:
[tex]\[ 2 - 5i, 2 - \sqrt{5} \][/tex]
Therefore, the complete list of zeros is:
[tex]\[ \boxed{2 - 5i, 2 - \sqrt{5}} \][/tex]
