Questions: Find a polynomial function of smallest degree with integer coefficients that has the given zeros. 1/2, 4-i, 4+i P(x)=

Solution Steps

To find a polynomial function of the smallest degree with integer coefficients that has the given zeros, we need to consider the following steps:

1. Identify the Zeros: The given zeros are \( \frac{1}{2}, 4-i, \) and \( 4+i \).
2. Conjugate Pairs: Since complex roots appear in conjugate pairs, \( 4-i \) and \( 4+i \) are already a conjugate pair.
3. Form Factors: The factors corresponding to these zeros are \( (x - \frac{1}{2}), (x - (4-i)), \) and \( (x - (4+i)) \).
4. Clear Fractions: To ensure integer coefficients, multiply the factor \( (x - \frac{1}{2}) \) by 2 to clear the fraction, resulting in \( (2x - 1) \).
5. Multiply Factors: Multiply all the factors to get the polynomial.

The given zeros of the polynomial are \( \frac{1}{2}, 4-i, \) and \( 4+i \).

The corresponding factors for these zeros are:

• For \( \frac{1}{2} \): \( 2x - 1 \)
• For \( 4-i \): \( x - (4 - i) = x - 4 + i \)
• For \( 4+i \): \( x - (4 + i) = x - 4 - i \)

We multiply the factors to obtain the polynomial: \[ P(x) = (2x - 1) \cdot (x - 4 + i) \cdot (x - 4 - i) \]

Upon expanding the polynomial, we find: \[ P(x) = 2x^3 - 17x^2 + 42x - 17 \]

Final Answer