Questions: Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros. 4+i1 2 The polynomial function in expanded form is f(x)=

Solution Steps

To find a polynomial function of the lowest degree with rational coefficients that has the given numbers as some of its zeros, we need to consider the complex conjugate pairs. If \(4 + i\) is a zero, then \(4 - i\) must also be a zero to ensure the polynomial has rational coefficients. We can then form the polynomial by multiplying the factors corresponding to these zeros.

Given the zero \(4 + i\), we recognize that its complex conjugate \(4 - i\) must also be a zero to ensure the polynomial has rational coefficients.

The polynomial can be formed using the zeros: \[ f(x) = (x - (4 + i))(x - (4 - i)) \]

Expanding the polynomial: \[ f(x) = (x - 4 - i)(x - 4 + i) = (x - 4)^2 - (i)^2 \] Since \(i^2 = -1\), we have: \[ f(x) = (x - 4)^2 + 1 \] Calculating \((x - 4)^2\): \[ f(x) = x^2 - 8x + 16 + 1 = x^2 - 8x + 17 \]

Final Answer