Unit 1 · Topic 1.5

1.5 Polynomial Functions and Complex Zeros

The zeros of a polynomial tell you its factors, its x-intercepts and where it is positive or negative. You'll count complex zeros, use conjugate pairs, read multiplicity from a graph, solve polynomial inequalities, and test whether a function is even or odd.

Key terms

complex zero
conjugate pair
multiplicity
linear factor
even and odd functions

Zeros and linear factors

If p(a) = 0, then a is a zero of p (also called a root of p(x) = 0). The number a can be real or a non-real complex number, like 2 + 3i, where i² = −1.

For a real number a, (x − a) is a factor of p exactly when a is a zero of p. So if p(4) = 0, then p(x) = (x − 4)·q(x) for some polynomial q.

If the factor (x − a) appears n times, the zero a has multiplicity n. Counting multiplicity, a polynomial of degree n has exactly n complex zeros. For example, (x − 1)²(x + 5) has degree 3 and zeros 1, 1 and −5.

Complex zeros come in conjugate pairs

For a polynomial with real coefficients (every polynomial in this course), non-real zeros come in pairs. If a + bi is a zero, so is its conjugate a − bi.

That's why a degree-3 polynomial with real coefficients must have at least one real zero: non-real zeros pair up, and 3 is odd.

Multiplying the pair gives a quadratic with real coefficients: (x − (a + bi))(x − (a − bi)) = (x − a)² + b².

Multiplicity and the graph

Each real zero a gives an x-intercept at (a, 0). How the graph behaves there depends on the multiplicity.

Odd multiplicity: the output changes sign at x = a, so the graph crosses the x-axis. With multiplicity 1 it crosses at an angle; with 3, 5, … it flattens out as it crosses.
Even multiplicity: the output has the same sign on both sides of a, so the graph touches the x-axis and turns back. It is tangent to the x-axis there.

Polynomial inequalities

To solve p(x) > 0, p(x) ≥ 0, p(x) < 0 or p(x) ≤ 0, find the real zeros first. They are the only places where p can change sign, so they are the endpoints of the solution intervals.

Then test the sign of p on each interval between zeros, or use multiplicity: the sign flips at odd-multiplicity zeros and stays the same at even ones. Include the zeros for ≥ or ≤, and leave them out for > or <.

Degree from a table

For outputs at equally spaced inputs, take successive differences. The degree of the polynomial is the smallest n for which the nth differences are constant. Constant first differences mean linear, constant second differences mean quadratic, constant third differences mean cubic, and so on.

Even and odd functions

A function is even if f(−x) = f(x) for every x in its domain. Its graph is symmetric over the y-axis. Example: x⁴ + 3x² − 2.

A function is odd if f(−x) = −f(x) for every x. Its graph is symmetric about the origin: rotating it 180° leaves it unchanged. Example: x³ − 5x.

A single power term axⁿ (n ≥ 1, a ≠ 0) is even if n is even and odd if n is odd. Most functions, like x³ + 1, are neither. To test, substitute −x and simplify.

Worked examples

Try each one yourself first, then open the solution.

Example 1
Finding all complex zeros
Find all zeros of p(x) = x³ − 2x² + 4x − 8.
Show the solutionHide the solution
1. Step 1: Group the terms: x²(x − 2) + 4(x − 2).
2. Step 2: Factor out (x − 2): p(x) = (x − 2)(x² + 4).
3. Step 3: Set each factor to zero. x − 2 = 0 gives x = 2. x² + 4 = 0 gives x² = −4, so x = ±2i.
4. Step 4: Check the count: degree 3, three zeros. The non-real zeros 2i and −2i are a conjugate pair, as expected.
Answer: x = 2, x = 2i and x = −2i. Only x = 2 is an x-intercept.
Example 2
Building a polynomial from its zeros
Write a polynomial with real coefficients, leading coefficient 1 and the least possible degree that has a zero of multiplicity 2 at x = 3 and a zero at x = 1 + i.
Show the solutionHide the solution
1. Step 1: Real coefficients force the conjugate 1 − i to be a zero too.
2. Step 2: Factors: (x − 3)² from the double zero, and (x − (1 + i))(x − (1 − i)) from the pair.
3. Step 3: Multiply the pair: (x − 1 − i)(x − 1 + i) = (x − 1)² − i² = x² − 2x + 1 + 1 = x² − 2x + 2.
4. Step 4: So p(x) = (x − 3)²(x² − 2x + 2), which has degree 4.
5. Step 5: Expanding: (x² − 6x + 9)(x² − 2x + 2) = x⁴ − 8x³ + 23x² − 30x + 18.
Answer: p(x) = (x − 3)²(x² − 2x + 2) = x⁴ − 8x³ + 23x² − 30x + 18.
Example 3
Trap: a double zero in an inequality
Solve (x + 2)(x − 1)²(x − 4) ≤ 0.
Show the solutionHide the solution
1. Step 1: Zeros: x = −2 (multiplicity 1), x = 1 (multiplicity 2), x = 4 (multiplicity 1).
2. Step 2: For x > 4, every factor is positive, so the product is positive.
3. Step 3: Crossing x = 4 (odd multiplicity) flips the sign: negative on 1 < x < 4.
4. Step 4: Crossing x = 1 (even multiplicity) does not flip the sign: still negative on −2 < x < 1. The trap is to assume the sign alternates at every zero.
5. Step 5: Crossing x = −2 flips the sign: positive for x < −2.
6. Step 6: The product is ≤ 0 where it is negative or zero: from −2 to 4, including the zeros.
Answer: −2 ≤ x ≤ 4, that is, the interval [−2, 4].

Common mistakes

Forgetting the conjugate. If 1 + i is a zero of a polynomial with real coefficients, 1 − i is a zero too, and it raises the degree.
Assuming signs alternate at every zero on a sign chart. At an even-multiplicity zero, the sign stays the same.
Testing even/odd by checking a single number. f(−x) = f(x) must hold for every x, so simplify f(−x) algebraically.
Thinking a function must be either even or odd. Most functions are neither.

On the exam

Expect to match a graph's x-intercepts with factors and multiplicities, or to choose a polynomial that matches a described graph.
Inequality questions reward a sign analysis organized by the zeros. Remember whether the endpoints are included.

Connected topics

Videos

Introduction to Complex Solutions of Polynomials (Precalculus - College algebra 35)
Professor LeonardWatch on YouTube (opens in a new tab)
AP Precalculus – 1.5B Even and Odd Polynomial Functions
The AlgebrosWatch on YouTube (opens in a new tab)
Even/Odd and Multiplicity in Under 3 mins (AP Precalculus Topic 1.5)
Maximum InsightWatch on YouTube (opens in a new tab)
1.5A - Polynomial Functions and Complex Zeros [AP Precalculus]
MrHelpfulNotHurtfulWatch on YouTube (opens in a new tab)
1.5-A Complex Zeros Video
Flamingo Math by Jean AdamsWatch on YouTube (opens in a new tab)
Even and Odd Functions - AP Precalculus Topic 1.5
Michael Porinchak - AP Statistics & AP PrecalculusWatch on YouTube (opens in a new tab)