Mathfolis

Polynomial Functions and Complex Zeros

Unit 1 · Polynomial and Rational Functions

What AP Precalc asks here

By the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n complex zeros counted with multiplicity. For polynomials with real coefficients, nonreal zeros come in conjugate pairs (a + bi and a − bi), so the count of real zeros has the same parity as the degree. The College Board uses Topic 1.5 to test counting nonreal zeros, constructing polynomials from a mixed set of real and complex roots, and reading multiplicity off a graph.

Counting zeros

Fundamental Theorem of Algebra
deg⁡p=(real zeros)+(nonreal complex zeros) (with multiplicity)\deg p = (\text{real zeros}) + (\text{nonreal complex zeros}) \text{ (with multiplicity)}
Conjugate Root Theorem
a+bi is a zero  ⟹  a−bi is also a zeroa + bi \text{ is a zero} \;\Longrightarrow\; a - bi \text{ is also a zero}

Graph behavior at a real zero of multiplicity k

k odd
graph crosses the x-axis at the zero\text{graph crosses the x-axis at the zero}
k even
graph touches the x-axis and bounces back\text{graph touches the x-axis and bounces back}
Caution: Don't count only the visible real zeros. A degree-5 polynomial with 3 real zeros has 2 nonreal zeros (one conjugate pair).
Type 1

Count complex zeros from degree and real zeros

Subtract the count of real zeros (with multiplicity) from the degree. Remaining zeros are nonreal and come in conjugate pairs.

Example 1
A real-coefficient polynomial has degree 6 and exactly 2 real zeros (each with multiplicity 1). How many nonreal complex zeros does it have? (A) 0 (B) 2 (C) 4 (D) 6

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Type 2

Construct a polynomial with specified zeros

If the polynomial must have real coefficients, every nonreal zero forces its conjugate. Multiply (x − r) factors for each zero (including conjugate pairs).

Example 2
Find the lowest-degree real-coefficient polynomial with zeros 33, −1-1, and 2+i2 + i. (A) (x−3)(x+1)(x−2−i)(x - 3)(x + 1)(x - 2 - i) (B) (x−3)(x+1)(x2−4x+5)(x - 3)(x + 1)(x^2 - 4x + 5) (C) (x−3)(x+1)(x2+4x+5)(x - 3)(x + 1)(x^2 + 4x + 5) (D) (x+3)(x−1)(x2−4x+5)(x + 3)(x - 1)(x^2 - 4x + 5)

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Type 3

Identify multiplicity from graph behavior

A graph that crosses the x-axis at x = r has odd multiplicity at r; one that touches and bounces has even multiplicity.

Example 3
A polynomial graph crosses the x-axis at x=2x = 2 and touches but bounces off at x=−1x = -1. What are the minimum multiplicities at x=2x = 2 and x=−1x = -1? (A) 1 at x=2x = 2, 1 at x=−1x = -1 (B) 1 at x=2x = 2, 2 at x=−1x = -1 (C) 2 at x=2x = 2, 1 at x=−1x = -1 (D) 2 at x=2x = 2, 2 at x=−1x = -1

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Polynomial Functions and Complex Zeros | AP Precalculus — Mathfolis