Question

a polynomial function has a root of 0 with multiplicity 1, and a root of 2 with multiplicity 4. if the function has a negative leading coefficient, and is of odd degree, which of the following are true?

• the function is positive on \\((-\infty, 0)\\).
• the function is negative on \\((0, 2)\\).
• the function is negative on \\((2, \infty)\\).
• the function is positive on \\((0, \infty)\\).

Explanation:

Construct the polynomial function

$$ f(x) = a \cdot x^1 \cdot (x - 2)^4, \quad a < 0 $$

Determine the sign of each interval

$$
LATEXBLOCK0
$$

Identify the true statements

$$
LATEXBLOCK1
$$

Answer:

• The function is positive on \((-\infty, 0)\). (Correct answer)
• The function is negative on \((0, 2)\). (Correct answer)
• The function is negative on \((2, \infty)\). (Correct answer)
• The function is positive on \((0, \infty)\).