Real and Complex Zeros of Polynomial Functions

different ways to find the zeros

Zeros of a polynomial function are the values of $x$ for which the polynomial equals zero: f(x)=0

Zeros correspond to:

• x-intercepts on the graph (when zeros are real)

• Roots or solutions of the polynomial equation

• Factors of the polynomial (via the Factor Theorem)

In Precalculus, understanding both real and complex zeros is essential for analyzing polynomial behavior, factoring polynomials, and sketching graphs.

Real Zeros

Real zeros are solutions that lie on the real number line. Graphically, a real zero corresponds to where the graph of the polynomial touches or crosses the x-axis.

Types of Real Zeros (by multiplicity)

A zero’s multiplicity describes how many times the factor appears.

If $$(x-a)^{n}$$ is a factor of f(x), then:

• a is a zero of multiplicity n

Rules:

• If n is odd, the graph crosses the x-axis.

• If n is even, the graph touches and turns around.

• Higher multiplicity → graph flattens near the zero.

Example:

$$f(x)=(x-2)^{2}(x+1)$$

• Zero at x=2, multiplicity 2 (touches axis)

• Zero at x=−1, multiplicity 1 (crosses axis)

Complex Zeros

Complex zeros appear when the polynomial has solutions that are not real numbers—typically when the discriminant of a quadratic is negative.

A complex zero has the form:

$$a+bi$$ where $$i=\sqrt{-1}$$​

Conjugate Pair Theorem

If a polynomial has real coefficients, then:

Complex zeros always occur in conjugate pairs.

That means:
If $$a+bi$$ is a zero, then $$a-bi$$ is also a zero.

Example:
If $$3+4i$$ is a zero, then $$3-4i$$ must also be a zero.

This rule is extremely important for factoring and for matching polynomials to their graphs.

Techniques for Finding Real Zeros

Factoring

Factor the polynomial and set each factor equal to zero.

Example:

$$f(x)=x^{3}-4x$$ factoring $$f(x)=x(x-2)(x+2)$$

Zeros: 0,2,−2

Rational Root Theorem

If f(x) has integer coefficients, any rational zero must be of the form: $$\frac{p}{q}$$​

where:

• p = factor of the constant term

• q = factor of the leading coefficient

Example:

f(x)=2×3+3×2−8f(x) = 2x^3 + 3x^2 – 8f(x)=2x3+3x2−8

Possible rational zeros:

$$\pm1,\pm2,\pm4,\pm8,\pm\frac{1}{2}$$

Test candidates using substitution or synthetic division.

Synthetic Division

Used to:

• Test candidate zeros

• Factor the polynomial after finding one zero

Example:
If x=3 is a zero, synthetic division reduces the polynomial’s degree by 1, simplifying further root-finding.

The Quadratic Formula

After factoring or dividing, many polynomials reduce to a quadratic:

$$ax^{2}+bx+c=0$$

Zeros:

x=$$\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$$​​

If the discriminant $$b^{2}-4ac$$ is:

• Positive → 2 real solutions

• Zero → 1 real solution (multiplicity 2)

• Negative → 2 complex conjugate solutions

Techniques for Finding Complex Zeros

Quadratic Formula (Negative Discriminant)

Example:

$$x^{2}+4x+13=0$$

x=$$\frac{-4\pm\sqrt{16-52}}{2}=-2\pm3i$$

Zeros:
x=$$-2\pm3i$$

Completing the Square

Useful for identifying complex roots and for rewriting in vertex form.

Using Conjugate Pairs to Build Polynomials

If zeros are $$3\pm2i$$, the polynomial must include:

$$(x-(3+2i))(x-(3-2i))$$

Multiply to form the real polynomial:

$$(x-3)^{2}+4=x^{2}+4x+13$$

Fundamental Theorem of Algebra

Every polynomial of degree n has exactly n complex roots (counting multiplicity).
This includes both real and non-real complex roots.

Examples:

• Degree 3 → exactly 3 total zeros

• Degree 5 → exactly 5 total zeros

A polynomial may have fewer real roots, but the total number including complex roots always matches the degree.

Graphical Interpretation

Real Zeros

Appear where the graph crosses or touches the x-axis.

Complex Zeros

Do not appear on the graph directly, but influence:

• Shape

• Turning points

• End behavior

Complete Analysis

Given:

$$f(x)=x^{4}+4x^{2}+13$$

Step 1: Try substitution
Let $$u=x^2$$ then

$$u^2+4u+13=0$$

Solve using quadratic formula:

$$u=-2\pm3i$$

Step 2: Convert back to $$x^{2}=u$$:

$$x^{2}=-2+3i$$ $$x^{2}=-2-3i$$

$$x=\pm\sqrt{-2+3i}$$ and $$x=\pm\sqrt{-2-3i}$$

These produce four complex roots, none real, consistent with degree 4.