Question

which of the following describes the zeroes of the graph of \\(f(x) = -x^5 + 9x^4 - 18x^3\\)?

\\(\bigcirc\\) 0 with multiplicity 3, -3 with multiplicity 2, and -2 with multiplicity 1
\\(\bigcirc\\) 0 with multiplicity 3, 3 with multiplicity 1, and 6 with multiplicity 1
\\(\bigcirc\\) 0 with multiplicity 3, 3 with multiplicity 2, and 2 with multiplicity 1
\\(\bigcirc\\) 0 with multiplicity 3, -3 with multiplicity 1, and -6 with multiplicity 1

Explanation:

Factor out the greatest common factor

Using the Polynomial Factoring knowledge point

$$ f(x) = -x^3(x^2 - 9x + 18) $$

Factor the remaining quadratic trinomial

Using the Polynomial Factoring knowledge point

$$ x^2 - 9x + 18 = (x - 3)(x - 6) $$

Write the fully factored polynomial

Using the Polynomial Factoring knowledge point

$$ f(x) = -x^3(x - 3)(x - 6) $$

Determine the zeroes and their multiplicities

Using the Polynomial Roots and Multiplicity of Roots knowledge points

$$
LATEXBLOCK0
$$

Answer:

• 0 with multiplicity 3, \(-3\) with multiplicity 2, and \(-2\) with multiplicity 1
• 0 with multiplicity 3, 3 with multiplicity 1, and 6 with multiplicity 1 (Correct answer)
• 0 with multiplicity 3, 3 with multiplicity 2, and 2 with multiplicity 1
• 0 with multiplicity 3, \(-3\) with multiplicity 1, and \(-6\) with multiplicity 1