Question

A "zero" of a function is an x-value where the function equals 0. For f(x) = -13x^2(5x+10)(x-3):

- Plug in x = 0: f(0) = -13·0^2·(5·0+10)·(0-3) = 0.
So x = 0 is a root, and the corresponding point on the graph is (0, 0).

Other roots:
- 5x + 10 = 0 → x = -2, so (-2, 0).
- x - 3 = 0 → x = 3, so (3, 0).

Note on multiplicity:
- x = 0 comes from the factor x^2, so it has multiplicity 2 (even). The graph touches the x-axis at (0,0) and bounces off rather than crossing.
- x = -2 and x = 3 have multiplicity 1, so the graph crosses the x-axis at those points.

I don't understand why (0, 0) is a zero explain like I'm 5

Answer 1

Imagine f(x) as a bunch of blocks you multiply together. If any one block is zero, the whole pile is zero — like if one box is empty, the whole stack is empty.

When x = 0:
- the factor x^2 becomes 0^2 = 0, so one of the blocks is zero.
- multiplying anything by 0 gives 0, so f(0) = 0.

Because x = 0 gives y = f(0) = 0, the point on the graph is (0, 0).

(Extra simple note: x^2 means there are two x-blocks, so the graph “bounces” at (0,0) instead of crossing.)