## Precalculus (6th Edition) Blitzer

The zeroes of the function are $1\,\text{ and }-1$ with the least possible multiplicity of two.
The zeros of the graph of a function are the points where the graph intersects the x-axis, i.e., the y-coordinate of that point is zero. Observe from the above graph that the function touches the x-axis at $x=1\,\text{ and }-1$. Thus, $1\,\text{ and }-1$ are the zeros of the function. This implies that $\left( x-1 \right)$ and $\left( x+1 \right)$ are the factors of the function. The multiplicity is defined as the exponent of the factors of the function, such that, r is the zero of the function. If r is of even multiplicity, then the graph will touch the x-axis and will turn around at r. If r is of odd multiplicity, then the graph of the function crosses the x-axis. Since the graph does not cross the x-axis, it instead turns around at the zeroes, and thus the zeroes have even multiplicities. Since the least even number is two, thus, $\left( x-1 \right)$ and $\left( x+1 \right)$ have at least two multiplicities. Therefore, the zeroes of the function are $1\,\text{ and }-1$ with the least possible multiplicity of two.