Find a polynomial equation with real coefficients that has the given zeros.
\[
3-5 i \text { and } 3+5 i
\]
The equation is $x^{2}-\square x+\square=0$
Answer
The polynomial equation with real coefficients that has the given zeros \(3-5i\) and \(3+5i\) is \(\boxed{x^{2}-6x+34=0}\).
Step 1 ：The roots of a polynomial equation with real coefficients always come in conjugate pairs if they are complex. This means that if \(3-5i\) is a root, then its conjugate, \(3+5i\), is also a root.
Step 2 ：We can use these roots to form the polynomial equation by setting it up as \((x - (3-5i))(x - (3+5i)) = 0\).
Step 3 ：Expanding this will give us the polynomial equation: \(x^{2} - 6x + 34\).
Step 4 ：The polynomial equation with real coefficients that has the given zeros \(3-5i\) and \(3+5i\) is \(\boxed{x^{2}-6x+34=0}\).
