Question

Which of the following could be the m u l t i p l i c i t y of zero whose graph touches the x-axis then turns back

Answer 1

To determine the multiplicity of a root (in this case, zero) based on the behavior of the graph at that root, we can use the following general guidelines:

1. **Odd Multiplicity**: If the graph crosses the x-axis at the root, the multiplicity is an odd number (1, 3, 5, ...).
2. **Even Multiplicity**: If the graph touches the x-axis and turns back, the multiplicity is an even number (0, 2, 4, ...).

Since you specified that the graph touches the x-axis and then turns back, this indicates that the root at zero has an even multiplicity.

Some possible multiplicities for the root at zero could therefore be:

- 0 (though this means zero is not a root),
- 2,
- 4,
- 6,
- etc.

So, the valid multiplicities of zero that satisfy your condition are any even positive integers.