Zeros of functions

In this chapter we focus on zeros (also called roots or x‑intercepts) of polynomial functions. We assume you already know what a polynomial function is and have some familiarity with its graph and degree from the parent chapter.

A zero of a function $f$ is a number $r$ such that
$$
f(r) = 0.
$$
For polynomial functions, zeros are extremely important because they describe where the graph crosses (or touches) the $x$‑axis and determine much of the function’s shape.

Zeros and factors

A central idea for polynomials is the connection between zeros and factors.

If $f(x)$ is a polynomial and $r$ is a number, then:

• $r$ is a zero of $f$
  $\iff$ $f(r) = 0$
  $\iff$ $(x - r)$ is a factor of $f(x)$.

So when you find a zero $r$, you immediately know that you can factor out $(x - r)$ from $f(x)$.

Example:

• Suppose $f(x) = x^2 - 5x + 6$.
• If $x = 2$ is a zero, then $f(2)=0$, and $(x-2)$ is a factor.
• If $x = 3$ is a zero, then $(x-3)$ is a factor.
• So
  $$
  f(x) = (x - 2)(x - 3).
  $$

Factored form and zeros

If a polynomial is written in factored form, its zeros are easy to read.

For example,
$$
f(x) = a(x - r_1)(x - r_2)\cdots(x - r_n),
$$
with $a \neq 0$, has zeros
$$
x = r_1,\ r_2,\ \dots,\ r_n.
$$

You find them by setting $f(x) = 0$ and using the fact that a product is zero only if at least one factor is zero:
$$
a(x - r_1)(x - r_2)\cdots(x - r_n) = 0
$$
implies
$$
x - r_1 = 0 \quad \text{or} \quad x - r_2 = 0 \quad \text{or} \quad \dots \quad \text{or} \quad x - r_n = 0.
$$

So:

• Each linear factor $(x - r_i)$ gives a zero $x = r_i$.

If a factor appears more than once, we get the idea of multiplicity.

Multiplicity of zeros

If a factor $(x - r)$ appears several times, we say that $r$ is a zero of that multiplicity.

• If $(x - r)$ appears once, $r$ is a simple zero (multiplicity 1).
• If $(x - r)^2$ is a factor, $r$ is a zero of multiplicity 2 (a double root).
• If $(x - r)^3$ is a factor, $r$ is a zero of multiplicity 3 (a triple root), and so on.

Formally, if
$$
f(x) = a(x - r)^m g(x),
$$
where $g(r) \neq 0$ and $m$ is a positive integer, then $r$ is a zero of multiplicity $m$.

Graphical effect of multiplicity

The multiplicity of a zero affects how the graph behaves at that $x$‑value:

• If the zero has odd multiplicity (1, 3, 5, …):
• The graph crosses the $x$‑axis at $x = r$.
• For multiplicity 1, it usually crosses in a “straight” manner.
• For higher odd multiplicity (like 3), it may flatten out slightly as it crosses.
• If the zero has even multiplicity (2, 4, 6, …):
• The graph touches the $x$‑axis at $x = r$ and turns around (it “bounces” off the axis).
• This is similar to the graph of $y = x^2$ touching the axis at $x = 0$.

Examples:

1. $f(x) = (x - 1)^2(x + 2)$
• Zero at $x = 1$ with multiplicity 2 (even): graph touches and turns at $x = 1$.
• Zero at $x = -2$ with multiplicity 1 (odd): graph crosses at $x = -2$.
2. $g(x) = (x + 3)^3$
• Zero at $x = -3$ with multiplicity 3 (odd): graph crosses, but flattens out near $x = -3$ (more like $x^3$).

Recognizing multiplicity from factored form is a quick way to predict this local behavior at the $x$‑axis.

Real zeros vs. complex zeros

Polynomial functions with real coefficients can have zeros that are:

• Real numbers (these correspond to $x$‑intercepts on the graph in the real plane).
• Complex numbers (of the form $a + bi$, where $b \neq 0$; these do not show up as $x$‑intercepts on a standard real graph).

When we talk about zeros of polynomial functions in an Algebra II context, we usually distinguish:

• Real zeros: solutions $x$ of $f(x) = 0$ that are real numbers.
• Nonreal complex zeros: complex solutions that are not real.

If the polynomial has real coefficients, then any nonreal complex zeros come in conjugate pairs:

• If $a + bi$ is a zero (with $b \neq 0$), then $a - bi$ is also a zero.

Example:

• $f(x) = x^2 + 1$.
• Solving $x^2 + 1 = 0$ gives $x^2 = -1$, so $x = i$ or $x = -i$.
• Both zeros are complex and not real; these do not appear as $x$‑intercepts on the real graph.

On the other hand:

• $g(x) = x^2 - 4$ has zeros $x = 2$ and $x = -2$, both real; they are visible $x$‑intercepts.

In this chapter we focus mainly on real zeros, because they are directly visible on the graph and closely tied to the real‑number behavior of the polynomial. Complex zeros are still zeros of the function, but they live in the complex plane.

Number of zeros and degree

The degree of a polynomial limits how many zeros it can have.

Let $f(x)$ be a polynomial of degree $n \ge 1$ (with real or complex coefficients). Then:

• Counting multiplicities and allowing complex zeros, $f$ has exactly $n$ zeros in the complex number system.
  (This is the Fundamental Theorem of Algebra; you only need an intuitive idea of it here.)
• The number of distinct real zeros is at most $n$.

Examples:

1. A linear polynomial (degree 1), like $f(x) = 3x - 7$, has exactly one zero (which is real, since its coefficient is nonzero and we can always solve for $x$).
2. A quadratic polynomial (degree 2), like $f(x) = ax^2 + bx + c$, has:
• 0, 1, or 2 real zeros.
• But always 2 zeros in the complex sense, counting multiplicity (for example, a repeated real root counts twice, or two complex conjugate roots).
3. A cubic polynomial (degree 3) has:
• At least one real zero, and up to 3 real zeros.
• Altogether 3 zeros in the complex number system, counting multiplicity.

You do not need all the theory here; in practice, remember:

• The maximum number of real zeros a polynomial can have is its degree.
• Any extra zeros beyond what you see on the real graph must be complex.

Graphs and zeros

For a polynomial function $f(x)$:

• A real zero $r$ corresponds to an $x$‑intercept of the graph, at the point $(r, 0)$.
• If you know the factorization of $f(x)$, you can mark all its real zeros on the $x$‑axis.
• Combined with the end behavior (from the degree and leading coefficient), and the multiplicity at each zero, you can sketch a fairly accurate graph.

To connect things:

1. From the algebraic form (especially factored form) of $f(x)$, you can:
• Identify zeros.
• Determine multiplicities.
• Predict whether each intercept is crossed or touched.
2. From the graph, you can:
• Estimate the $x$‑values where $f(x) = 0$.
• Guess the multiplicity (roughly) from whether the graph crosses or bounces at the $x$‑axis.

These two viewpoints—algebraic and graphical—describe the same structure: the zeros of the polynomial.

Summary of key ideas

• A zero of a function $f$ is a number $r$ such that $f(r) = 0$.
• For polynomials, $r$ is a zero $\iff$ $(x - r)$ is a factor.
• If $f(x)$ is written as $a(x - r_1)(x - r_2)\cdots(x - r_n)$, then its zeros are $x = r_1, r_2, \dots, r_n$.
• The multiplicity of a zero is how many times its factor appears:
• Odd multiplicity: graph crosses the $x$‑axis at that zero.
• Even multiplicity: graph touches and turns at that zero.
• A polynomial of degree $n$ has at most $n$ real zeros, and exactly $n$ zeros in the complex number system, counting multiplicities.
• Real zeros correspond to $x$‑intercepts of the graph; nonreal complex zeros do not appear on the real $xy$‑plane graph.

Later chapters on factoring, quadratic equations, and more advanced polynomial techniques will give you systematic methods to find these zeros for specific polynomials. Here the focus is on understanding what zeros are, how they are related to factors, and what they mean for the graph of a polynomial function.