Question

find a polynomial function of least degree having only real coefficients, a leading coefficient of 1, and roots of (1 - sqrt{3}), (1 + sqrt{3}), and (4 - i).
the polynomial function is (p(x) = square).
(simplify your answer.)

Explanation:

Step1: Identify conjugate roots

For a polynomial with real coefficients, complex roots come in conjugate pairs. Given root \(4 - i\), its conjugate \(4 + i\) is also a root. The other roots are \(1 - \sqrt{3}\) and \(1 + \sqrt{3}\).

Step2: Form factors from roots

The factors of the polynomial are \((x - (1 - \sqrt{3}))\), \((x - (1 + \sqrt{3}))\), \((x - (4 - i))\), and \((x - (4 + i))\).

Step3: Multiply factors with radical roots

First, multiply \((x - (1 - \sqrt{3}))(x - (1 + \sqrt{3}))\):

$$
LATEXBLOCK0
$$

Step4: Multiply factors with complex roots

Next, multiply \((x - (4 - i))(x - (4 + i))\):

$$
LATEXBLOCK1
$$

Step5: Multiply the two quadratic polynomials

Now, multiply \( (x^2 - 2x - 2)(x^2 - 8x + 17) \):

$$
LATEXBLOCK2
$$

Answer:

\(P(x) = x^4 - 10x^3 + 31x^2 - 18x - 34\)