Lesson 2.15

Finding All Zeros

A degree- polynomial has exactly zeros (counting multiplicity and complex numbers). Now we combine every tool — Rational Root Theorem, synthetic division, and the Quadratic Formula — to find them all.

Introduction

Until now, you've found rational zeros. But many polynomials have irrational or imaginary zeros too. This lesson teaches you a systematic workflow to find every last zero of any polynomial.

Past Knowledge

Rational Root Theorem, Factor Theorem, synthetic division, and the Quadratic Formula.

Today's Goal

Find all zeros — real and imaginary — by reducing degree until a quadratic remains.

Future Success

Knowing all zeros lets you sketch the full graph and is essential for partial fractions in calculus.

Key Concepts

Fundamental Theorem of Algebra

A polynomial of degree has exactly zeros

(counting multiplicity, over the complex numbers)

Complex Conjugate Pairs Theorem

If a polynomial has real coefficients and is a zero, then is also a zero. Imaginary roots always come in conjugate pairs.

The Master Workflow

1
Use the Rational Root Theorem to list candidates
2
Test candidates with synthetic division
3
Each success reduces the degree by 1
4
Repeat until you reach a quadratic
5
Use the Quadratic Formula on the remaining quadratic

Worked Examples

Example 1: All Rational Zeros

Basic

Find all zeros of .

1

Rational Root Candidates

Candidates:

2

Test via synthetic division

−112−5−6
−1−16
11−60

✓ Quotient:

3

Factor the Remaining Quadratic

All zeros:

Example 2: Mix of Rational and Irrational Zeros

Intermediate

Find all zeros of .

1

Test

✓

2

Synthetic Division by 1

11−3−24
1−2−4
1−2−40

Quotient:

3

Quadratic Formula on

All zeros:

(1 rational + 2 irrational = 3 zeros for a degree-3 poly ✓)

Example 3: Complex (Imaginary) Zeros

Advanced

Find all zeros of .

1

Factor by Grouping (or test )

2

Solve

✓

Final Answer

All zeros:

(1 real + 2 imaginary conjugates = 3 zeros ✓)

Common Pitfalls

Forgetting Imaginary Zeros

A degree-3 polynomial must have 3 zeros. If you only find 1 real root and claim you're done, you're missing the conjugate pair from the remaining quadratic.

Using the Wrong Quotient

After synthetic division, the quotient has degree . Don't forget: the bottom row gives the coefficients of the quotient, not the roots.

Real-Life Applications

Electrical engineers analyze circuits using polynomials called characteristic equations. The zeros — including complex ones — determine whether a circuit oscillates, dampens, or amplifies. Complex zeros produce oscillatory behavior (like radio waves), while real zeros produce smooth exponential responses.

Practice Quiz

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