1.5 Polynomial Functions and Complex Zeros

A complex number $a$ is a zero of a polynomial $p$ when $p(a) = 0$, and a real zero $a$ means $(x - a)$ is a factor and the graph has an x-intercept at $(a, 0)$. A polynomial of degree $n$ has exactly $n$ complex zeros counting multiplicities. Non-real zeros come in conjugate pairs, and the multiplicity of a real zero tells you whether the graph crosses the x-axis or just touches it.

Why This Matters for the AP Precalculus Exam

Zeros are the backbone of working with polynomials. Once you know the zeros and their multiplicities, you can sketch graphs, write factored forms, and solve polynomial inequalities by using the zeros as endpoints. On the AP Precalculus exam, you may need to find zeros by hand when a polynomial factors cleanly, and you may use a graphing calculator to find zeros on questions that allow technology. Free-response questions often ask you to explain your reasoning, so being able to connect a zero, its factor, and its graph behavior in words is useful for clear exam work.

Recognizing even and odd symmetry helps you predict graph behavior quickly and check whether an equation matches a graph, which shows up in multiple-choice questions about representations.

Key Takeaways

• A complex number a is a zero of p when p(a) = 0; for a real zero, (x - a) is a linear factor.
• A degree n polynomial has exactly n complex zeros when you count multiplicities.
• Non-real zeros come in conjugate pairs: if a + bi is a zero, so is a - bi.
• Odd multiplicity means the graph crosses the x-axis; even multiplicity means it is tangent and touches without crossing.
• Real zeros become x-intercepts at (a, 0) and serve as endpoints when solving polynomial inequalities.
• An even function satisfies f(-x) = f(x) (symmetry over the y-axis); an odd function satisfies f(-x) = -f(x) (symmetry about the origin).

Real, Imaginary, and Complex Numbers

Before working with zeros, it helps to be clear on the kinds of numbers involved.

Real Numbers

Real numbers can be placed as a single point on the number line. They include integers, fractions, decimals, and roots, and they can be positive, negative, or zero.

Examples:

• 2
• 3.14
• 1/2
• √5 (an irrational number)

Imaginary Numbers

Imaginary numbers are the product of a real number and the imaginary unit i, where i is defined as the square root of -1. They are usually written as bi, where b is a real number.

Examples:

• 2i
• 3.5i
• (1/2)i

Complex Numbers

Complex numbers are written as a + bi, a sum of a real part a and an imaginary part bi. Every real number and every imaginary number is also a complex number.

Examples:

• 3 + 4i
• 2i
• 1/2 + 2i
• √3 - i

Linear Factors and Multiplicities

A complex number a is a zero of a polynomial function p if p(a) = 0. In other words, a is a root of the equation p(x) = 0. If a is a real number, then (x - a) is a linear factor of p if and only if a is a zero of p.

For polynomials with only real coefficients, non-real zeros always show up in conjugate pairs. If a + bi is a zero, then a - bi must also be a zero.

If a polynomial has a repeated linear factor (x - a) raised to the power n, then a is a zero of multiplicity n. When you factor the polynomial, the factor (x - a) appears n times.

A polynomial function of degree n has exactly n complex zeros when counting multiplicities. So if a zero has multiplicity m, it is counted m times in that total.

X-Intercepts

If p has a real zero at a, the graph of p crosses or touches the x-axis at the point (a, 0), because p(a) = 0.

This is why real zeros are useful for solving polynomial inequalities. The zeros split the number line into intervals, and the sign of p stays the same across each interval between consecutive zeros. You can pick one test point in each interval to decide whether p is positive or negative there, then use the real zeros as endpoints to describe where p(x) > 0 or p(x) < 0.

Even Multiplicities

When a real zero a has an even multiplicity, the factor (x - a) appears an even number of times. Near x = a, the output values keep the same sign on both sides, so the graph is tangent to the x-axis at x = a. The graph touches the axis and turns around, staying on the same side instead of crossing.

Odd Multiplicities

When a real zero a has an odd multiplicity, the output values change sign as you pass through x = a. The graph crosses the x-axis at that point.

Working with Polynomial Functions

Determining Degree Using Successive Differences

To find the degree of a polynomial from a table of equally spaced input values, look at the pattern of differences between output values. Compute the first differences, then the second differences, and keep going until some nth set of differences is constant.

The degree equals the smallest n for which the nth differences are constant. This connects back to rates of change: constant first differences mean a steady rate, so the function is linear; constant second differences mean the rate of change itself changes steadily, so the function is quadratic, and so on.

• Constant first differences: degree 1 (linear)
• Constant second differences: degree 2 (quadratic)
• Constant third differences: degree 3 (cubic)

This is called the method of successive differences, and it works for any polynomial.

Finding the Degree Practice Question

Suppose we have the polynomial function $p(x) = 2x^3 - 5x^2 + 3x + 1$. We want to find its degree using successive differences.

First, compute the first differences:

$p(x) = 2x^3 - 5x^2 + 3x + 1$

$p(x+1) = 2(x+1)^3 - 5(x+1)^2 + 3(x+1) + 1$

$p(x+1) - p(x) = 6x^2 - 4x$

The first differences are not constant, so compute the second differences:

$6x^2 - 4x$

$(6(x+1)^2 - 4(x+1)) - (6x^2 - 4x) = 12x + 2$

Still not constant, so compute the third differences:

$12x + 2$

$(12(x+1) + 2) - (12x + 2) = 12$

The third differences are constant at 12. So the degree of p(x) is 3, since 3 is the smallest n for which the nth differences are constant. The function is cubic.

Even Functions

An even function has two key properties:

1. It is symmetric over the line x = 0, so reflecting the graph across the y-axis gives back the same graph.
2. It satisfies f(-x) = f(x), so replacing x with -x gives the same output.

For a polynomial of the form $p(x) = a_nx^n$ where n is even and $a_n \neq 0$, the function is even, because:

$p(-x) = a_n(-x)^n = a_nx^n = p(x)$

Odd Functions

An odd function also has two key properties:

1. It is rotationally symmetric about the point (0, 0), so rotating the graph 180 degrees about the origin gives back the same graph.
2. It satisfies f(-x) = -f(x), so replacing x with -x flips the sign of the output.

For a polynomial of the form $p(x) = a_nx^n$ where n is odd and $a_n \neq 0$, the function is odd, because:

$p(-x) = a_n(-x)^n = -a_nx^n = -p(x)$

How to Use This on the AP Precalculus Exam

Problem Solving

• When a polynomial factors cleanly, set each factor equal to zero to find real zeros, and read off the multiplicity from the exponent on each factor.
• Use the factor and zero connection both ways: a real zero a gives the factor (x - a), and the factor (x - a) gives the zero a.
• For non-real zeros, remember they travel in conjugate pairs. If you are told a + bi is a zero, you automatically know a - bi is also a zero.
• Match degree to the number of zeros: a degree n polynomial has n complex zeros counting multiplicities, which helps you check whether you have found all of them.

Graphing and Representations

• Use multiplicity to decide graph behavior at each x-intercept: odd multiplicity crosses, even multiplicity is tangent and turns around.
• Check for symmetry to match equations to graphs. Test f(-x): if it equals f(x), the function is even; if it equals -f(x), the function is odd.

Calculator Use

• On questions that allow technology, you can find real zeros, points of intersection, and extrema with a graphing calculator. Use this when a polynomial does not factor by hand.

Common Trap

• Show your reasoning on free-response questions. Connecting a zero, its factor, and its graph behavior in clear words is important for clear exam work.

Common Misconceptions

• A degree n polynomial has n complex zeros counting multiplicities, not always n different x-intercepts. Repeated zeros and non-real zeros reduce the number of distinct points where the graph meets the x-axis.
• Even multiplicity does not mean the graph crosses the x-axis. It is tangent there and stays on the same side. Odd multiplicity is the case that crosses.
• Non-real zeros only have to come in conjugate pairs when the polynomial has real coefficients. Do not assume conjugate pairs for polynomials with non-real coefficients.
• An even function is not the same as a polynomial with even degree in general. The clean rule f(-x) = f(x) holds for single-term powers like $a_nx^n$ with n even; for full polynomials, check f(-x) directly.
• A zero of the polynomial is an input value where p(a) = 0, not the point (a, 0) itself. The point (a, 0) is the x-intercept that matches a real zero.
• Successive differences only reveal the degree when the input values are equally spaced. Uneven spacing breaks the pattern.

Related AP Precalculus Guides

• 1.9 Rational Functions and Vertical Asymptotes
• 1.11 Equivalent Representations of Polynomial and Rational Expressions
• 1.4 Polynomial Functions and Rates of Change
• 1.8 Rational Functions and Zeros
• 1.12 Transformations of Functions
• 1.3 Rates of Change in Linear and Quadratic Functions