Precalculus / Unit 2: Polynomial and Rational Functions
2.4 Rational and Complex Zeros
Notes
- Rational Root Theorem: any rational zero of a polynomial with integer coefficients is $\displaystyle \frac{p}{q}$, where $p$ divides the constant term and $q$ divides the leading coefficient.
- Fundamental Theorem of Algebra: a degree $n$ polynomial has exactly $n$ complex zeros, counting multiplicity.
- Conjugate Pairs Theorem: if a polynomial has real coefficients and $a + bi$ is a zero, then $a - bi$ is also a zero.
- Example: $x^2 + 4 = 0$ gives $x = \pm 2i$.
Practice
Which number is NOT a possible rational zero of $2x^3 - 3x^2 - 11x + 6$?
$p$ must divide 6 and $q$ must divide 2. The denominator 3 does not divide 2.
A polynomial with real coefficients has $2 + 3i$ as a zero. Which must also be a zero?
Complex zeros come in conjugate pairs.
How many zeros (counting multiplicity and complex zeros) does a degree 4 polynomial have?
The Fundamental Theorem of Algebra gives exactly $n$ zeros for degree $n$.
Ready for more? Take the Unit 2 test.