Question

which is the polynomial function of lowest degree with rational real coefficients, a leading coefficient of 3 and roots \\(\sqrt{5}\\) and 2?

\\(f(x) = 3x^3 - 6x^2 - 15x + 30\\)
\\(f(x) = x^3 - 2x^2 - 5x + 10\\)
\\(f(x) = 3x^2 - 21x + 30\\)
\\(f(x) = x^2 - 7x + 10\\)

Explanation:

Identify the required roots and apply the Conjugate Root Theorem

Using the Polynomial Roots knowledge point
Since the polynomial must have rational coefficients, any irrational root of the form \(a + \sqrt{b}\) must have its conjugate \(a - \sqrt{b}\) as a root.
Given root: \(\sqrt{5}\) (which is \(0 + \sqrt{5}\))
Conjugate root: \(-\sqrt{5}\)
Other given root: \(2\)
Therefore, the roots of the polynomial are \(\sqrt{5}\), \(-\sqrt{5}\), and \(2\).

Write the polynomial in factored form

Using the Polynomial Factoring knowledge point

$$ f(x) = a(x - \sqrt{5})(x + \sqrt{5})(x - 2) $$

where \(a\) is the leading coefficient.

Expand the factored form

Using the Polynomial Factoring knowledge point

$$
LATEXBLOCK0
$$

Apply the leading coefficient

Using the Polynomial Roots knowledge point
Given leading coefficient: \(a = 3\)

$$
LATEXBLOCK1
$$

Answer:

• (A) \(f(x) = 3x^3 - 6x^2 - 15x + 30\) (Correct answer)
• (B) \(f(x) = x^3 - 2x^2 - 5x + 10\)
• (C) \(f(x) = 3x^2 - 21x + 30\)
• (D) \(f(x) = x^2 - 7x + 10\)