Algebra 2 / Unit 3: Polynomial Functions
3.5 Zeros and the Fundamental Theorem of Algebra
Notes
- Fundamental Theorem of Algebra: a polynomial of degree $n \ge 1$ has exactly $n$ complex zeros, counting repeats.
- If the coefficients are real, nonreal zeros come in conjugate pairs: if $2 + i$ is a zero, so is $2 - i$.
- Rational Root Theorem: any rational zero is $\displaystyle \pm\frac{\text{factor of constant}}{\text{factor of leading coefficient}}$.
- Factor by grouping when you can: $x^3 + 2x^2 - 9x - 18 = x^2(x + 2) - 9(x + 2) = (x + 2)(x - 3)(x + 3)$.
Practice
How many complex zeros, counting multiplicity, does a degree-6 polynomial have?
The Fundamental Theorem of Algebra says a degree-$n$ polynomial has $n$ zeros.
A polynomial with real coefficients has zeros 2 and $3 + i$. Which must also be a zero?
Nonreal zeros of real polynomials come in conjugate pairs.
Find all zeros of $x^3 - x^2 - 4x + 4$.
Group: $x^2(x - 1) - 4(x - 1) = (x - 1)(x - 2)(x + 2)$.
Ready for more? Take the Unit 3 test.