Integrated Math 3 / Unit 1: Polynomial Functions
1.2 Zeros and Multiplicity
Notes
A zero (root) of $f$ is an $x$-value where $f(x) = 0$. On a graph it is an $x$-intercept. If $(x - r)^m$ is a factor, then $r$ is a zero of multiplicity $m$.
- Odd multiplicity: the graph crosses the $x$-axis at $r$ (for multiplicity 3 or more it flattens as it crosses).
- Even multiplicity: the graph touches the axis at $r$ and turns around.
- The multiplicities add up to the degree (counting complex zeros).
- Example: $f(x) = (x - 3)^2 (x + 1)^3$ has a zero at 3 (multiplicity 2, touches) and a zero at $-1$ (multiplicity 3, crosses). The degree is 5.
Practice
What happens to the graph of $f(x) = x(x - 2)^2$ at $x = 2$?
The factor $(x - 2)^2$ has even multiplicity, so the graph touches the axis and turns.
What is the lowest-degree polynomial with a zero at 2 of multiplicity 2 and a zero at $-3$ of multiplicity 1 (leading coefficient 1)?
Zero $r$ gives the factor $(x - r)$. So 2 gives $(x - 2)^2$ and $-3$ gives $(x + 3)$.
How many zeros, counting multiplicity, does $f(x) = x^2 (x - 5)^3 (x + 1)$ have?
The multiplicities are $2 + 3 + 1 = 6$, which equals the degree.
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