Final answer:
The polynomial function of lowest degree with rational real coefficients, a leading coefficient of 3, and roots √5 and 2 is f(x) = 3x³ - 6x² - 15x + 30.
Step-by-step explanation:
The polynomial function of lowest degree with rational real coefficients, a leading coefficient of 3, and roots √5 and 2 is:
f(x) = 3x³ - 6x² - 15x + 30
To find the polynomial function, we use the roots and the leading coefficient. Since the roots are √5 and 2, we know that the factors of the polynomial are (x - √5) and (x - 2). Multiplying these factors together, we get:
(x - √5)(x - 2) = x² - 2x - √5x + 2√5 = x² - (2 + √5)x + 2√5x - 2√5 = x² - (2 + √5)x + 2√5(x - 1)
Simplifying further, we have:
x² - (2 + √5)x + 2√5(x - 1) = x² + (2√5 - 2 - √5)x + 2√5(x - 1) = x² - (2 - √5)x + 2√5(x - 1)
Finally, multiplying by the leading coefficient 3, we obtain:
f(x) = 3x³ - (6 - 3√5)x² + 6√5(x - 1)