AP Precalculus / Unit 1: Polynomial and Rational Functions
1.5 Polynomial Functions and Complex Zeros
Notes
A degree-$n$ polynomial has exactly $n$ complex zeros, counting multiplicity.
- If $a$ is a real zero, $(x-a)$ is a factor and $(a,0)$ is an $x$-intercept.
- Even multiplicity: the graph touches and turns around. Odd multiplicity: it crosses.
- Non-real zeros come in conjugate pairs $a\pm bi$.
Practice
$p(x)=(x-2)^2(x+1)$. At $x=2$ the graph:
Multiplicity 2 is even.
A degree-4 polynomial with real coefficients has zeros $3$, $-1$ and $2+i$. The fourth zero is:
Conjugate pairs.
How many complex zeros does $x^6-1$ have?
Degree 6.
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