1.5 Polynomial Functions and Complex Zeros

Complex zeros come in pairs

For polynomials with real coefficients, complex zeros always occur in conjugate pairs: if a+bi is a zero, so is a−bi. This means non-real zeros always come in twos, so a polynomial's non-real zero count is always even.

Fundamental Theorem of Algebra

A polynomial of degree n has exactly n zeros counted with multiplicity, allowing for complex zeros. This guarantees every polynomial can be fully factored into n linear factors over the complex numbers, even if some factors involve imaginary numbers.

Linear factorization & multiplicity

Every polynomial can be written as a product of linear factors: f(x) = a(x−r₁)(x−r₂)...(x−rₙ). A zero's multiplicity is how many times its factor repeats. Real zeros with odd multiplicity cross the x-axis; even multiplicity means the graph touches and turns at that zero.

Key terms

Fundamental Theorem of Algebra

A polynomial of degree n has exactly n zeros, counted with multiplicity, over the complex numbers.

Complex conjugate pair

For real-coefficient polynomials, if a+bi is a zero, then a−bi must also be a zero.

Multiplicity

The number of times a given zero's linear factor appears in the factorization of a polynomial.

Odd multiplicity behavior

The graph crosses the x-axis at that zero.

Even multiplicity behavior

The graph touches the x-axis at that zero and turns back without crossing.

Try an AP-style question

How does the graph of p(x) = (x − 2)²(x + 1) behave at x = 2?

1. It crosses the x-axis
2. It has a vertical asymptote
3. It touches the x-axis and turns around
4. It has a hole