Find a polynomial function with real coefficients that has the given zeros.1, 5i

1 Answers

Best Answer

Since 1 and 5i are zeros, the function will have (x-1) and (x-5i) as factors. Complex zeros come in conjugate pairs so (x+5i) is also a factor.
The polynomial function can be written as f(x)=a(x-1)(x-5i)(x+5i), where ain mathbb(R).
Let us substitute 1 for a and multiply to get one of the functions:
f(x)=(x-1)(x-5i)(x+5i)
=(x-1)(x^(2)-5ix+5ix-25i^(2))
=(x-1)(x^(2)+25)
Let us multiply terms in the last two pairs of parentheses:
=x^(3)+25x-x^(2)-25
=x^(3)-x^(2)+25x-25
Result:
x^(3)-x^(2)+25x-25