Question

what is the polynomial function of lowest degree with lead coefficient 1 and roots \\(i\\), \\(-2\\), and \\(2\\)?

\\(f(x) = x^3 - x^2 - 4x + 4\\)
\\(f(x) = x^4 - 3x^2 - 4\\)
\\(f(x) = x^4 + 3x^2 - 4\\)
\\(f(x) = x^3 + x^2 - 4x - 4\\)

Explanation:

Identify all roots of the polynomial

Using the Roots of Polynomials knowledge point
Since the polynomial has real coefficients, the complex root \(i\) must have its conjugate \(-i\) as a root.
Thus, the complete set of roots is:

$$ x = i, \quad x = -i, \quad x = -2, \quad x = 2 $$

Set up the linear factors

Using the Linear Factorization Theorem knowledge point

$$ f(x) = (x - i)(x + i)(x + 2)(x - 2) $$

Multiply the factors

Using the Polynomial Multiplication knowledge point

$$
LATEXBLOCK0
$$

Answer:

• (A) \(f(x) = x^3 - x^2 - 4x + 4\)
• (B) \(f(x) = x^4 - 3x^2 - 4\) (Correct answer)
• (C) \(f(x) = x^4 + 3x^2 - 4\)
• (D) \(f(x) = x^3 + x^2 - 4x - 4\)