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Fundamental theorem of algebra

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Algebra II
22. CC.HSA.APR.D.6
22.3 Fundamental theorem of algebra

Fundamental Theorem of Algebra

See why every polynomial of degree n has exactly n complex zeros, and learn to find and count them with clear worked examples.

What You'll Learn

State the fundamental theorem of algebra and explain what "at least one complex zero" guarantees.
Apply the linear factorization corollary to write a degree n polynomial as n linear factors.
Count zeros correctly by including multiplicity and non-real complex zeros, not just real roots.
Recognize that complex zeros of real-coefficient polynomials always come in conjugate pairs.
Solve for all zeros of a given polynomial using factoring and the quadratic formula.
Connect the theorem to the factor theorem and rational zero test to fully factor polynomials.

What You'll Practice

1

Finding all roots of degree 3 and higher polynomials using factoring techniques

2

Applying the quadratic formula to irreducible quadratic factors

3

Counting real and imaginary roots including multiplicity

4

Listing possible root combinations for polynomials of varying degrees

Why This Matters

The fundamental theorem of algebra is essential for understanding polynomial behavior in calculus, engineering, and physics. It guarantees that every polynomial equation has solutions and helps you predict how many x-intercepts a function will have, which is critical for graphing, optimization, and solving real-world problems.

Before You Start — Make Sure You Can:

Factoring perfect square trinomials: (a + b)^2 = a^2 + 2ab + b^2 or (a - b)^2 = a^2 - 2ab + b^2Factoring trinomialsPolynomial synthetic division

This Unit Includes

3 Video lessons
Practice exercises
Learning resources

Skills

Polynomial Roots
Fundamental Theorem of Algebra
Imaginary Numbers
Factoring
Multiplicity
Quadratic Formula
Irreducible Quadratics
Complex Conjugates
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