Precalculus / Unit 2: Polynomial and Rational Functions
2.2 Zeros and Multiplicity
Notes
A zero is an x-value where $f(x) = 0$. If $(x - r)^m$ is a factor, then $r$ is a zero of multiplicity $m$.
- Odd multiplicity: the graph crosses the x-axis at $r$.
- Even multiplicity: the graph touches the axis and turns around.
- The y-intercept is $f(0)$.
- Example: $f(x) = (x - 2)^2 (x + 3)^3$ has degree 5. It touches at $x = 2$ and crosses at $x = -3$.
Practice
For $f(x) = (x - 2)^2 (x + 3)^3$, what happens at $x = 2$?
The multiplicity of 2 is even.
What are the zeros of $f(x) = x^3 - 4x$?
Factor: $x(x^2 - 4) = x(x - 2)(x + 2)$.
What is the y-intercept of $f(x) = (x - 2)(x + 1)(x - 3)$?
$f(0) = (-2)(1)(-3) = 6$.
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