Which of the following describes the zeroes of the graph of f(x) = -x⁵ + 9x⁴ - 18x³?
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
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

Solution:

It is given that

f(x) = -x⁵ + 9x⁴ - 18x³

Taking out x³ as common

f(x) = -x³ (x² - 9x + 18)

Using the quadratic formula x = [-b ± √(b² - 4ac)]/2a

Substituting the values

x = [-(-9) ± √((-9)² - 4(1)(18))]/2(1)

x = [9 ± √(81 - 72)]/ 2

x = [9 ± √9]/2

x = (9 ± 3)/2

Here

x = (9 + 3)/2 and x = (9 - 3)/2

x = 12/12 and x = 6/2

x = 6 and x = 3

So we get

f(x) = -x³ (x - 3) (x - 6)

So the zeros of the function are

x = 0 with multiplicity 3

x = 3 with multiplicity 1

x = 6 with multiplicity 1

Therefore, 0 with multiplicity 3, 3 with multiplicity 1, and 6 with multiplicity 1 describe the zeroes of the graph.

---

Which of the following describes the zeroes of the graph of f(x) = -x⁵ + 9x⁴ - 18x³?
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
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

Summary:

0 with multiplicity 3, 3 with multiplicity 1, and 6 with multiplicity 1 describe the zeroes of the graph f(x) = -x⁵ + 9x⁴ - 18x³.