Question

Find the zeros of the function, and then describe the behavior of the graph at each zero. f(x) = x³ - 2x²-9x+18

The zero(s) is/are (Use a comma to separate answers as needed.)​

Answer 1

The zeros of
`\(f(x) = x^3 - 2x^2 - 9x + 18\)` are x = 3, -3, 2. At x = 3, the graph touches; at -3, it crosses; at 2, it crosses.

To find the zeros of the function
`\(f(x) = x^3 - 2x^2 - 9x + 18\)`, set f(x) equal to zero and solve for x:

`\[x^3 - 2x^2 - 9x + 18 = 0\]`

One zero is x = 3. Perform polynomial division to factor out (x - 3) from the cubic expression:

`\[x^3 - 2x^2 - 9x + 18 = (x - 3)(x^2 + x - 6)\]`

Factorizing further, we get (x - 3)(x + 3)(x - 2). Therefore, the zeros are
x = 3, -3, 2.

At x = 3, the graph touches the x-axis. At x = -3, it crosses the x-axis, and at x = 2, it also crosses, indicating the behavior of a root, a root with multiplicity, and a simple root, respectively.