Find a polynomial function of least degree having only real coefficients, a leading coefficient of 1 , and zeros of \( 4,1-2 i \), and \( 4+2 i \).

To construct a polynomial of the least degree with the given specifications, we need to consider the following: 1. **Given Zeros:** - Real zero: \(4\) - Complex zeros: \(1 - 2i\) and \(4 + 2i\) 2. **Complex Conjugate Zeros:** - For a polynomial with real coefficients, complex zeros must come in conjugate pairs. Therefore, the conjugates of the given complex zeros must also be included: - Conjugate of \(1 - 2i\): \(1 + 2i\) - Conjugate of \(4 + 2i\): \(4 - 2i\) 3. **Total Zeros:** - Real zero: \(4\) - Complex zeros: \(1 - 2i, 1 + 2i, 4 + 2i, 4 - 2i\) 4. **Constructing the Polynomial:** - Each zero \(\alpha\) corresponds to a factor \((x - \alpha)\) in the polynomial. - Therefore, the polynomial is: \[ P(x) = (x - 4)(x - (1 - 2i))(x - (1 + 2i))(x - (4 + 2i))(x - (4 - 2i)) \] 5. **Simplifying the Polynomial:** - First, simplify the pairs of complex conjugate factors: \[ (x - (1 - 2i))(x - (1 + 2i)) = (x - 1)^2 + (2)^2 = x^2 - 2x + 5 \] \[ (x - (4 + 2i))(x - (4 - 2i)) = (x - 4)^2 + (2)^2 = x^2 - 8x + 20 \] - Now, multiply all the factors together: \[ P(x) = (x - 4)(x^2 - 2x + 5)(x^2 - 8x + 20) \] - Expanding further: \[ P(x) = (x - 4)(x^4 - 10x^3 + 41x^2 - 80x + 100) \] \[ P(x) = x^5 - 14x^4 + 81x^3 - 244x^2 + 420x - 400 \] 6. **Final Polynomial:** \[ P(x) = x^5 - 14x^4 + 81x^3 - 244x^2 + 420x - 400 \] This is the polynomial of least degree with real coefficients, a leading coefficient of \(1\), and the given zeros. **Answer:** After expanding, one such polynomial is \(x^5 - 14\,x^4 + 81\,x^3 - 244\,x^2 + 420\,x - 400\).