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Fundamental Theorem of Algebra

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Fundamental Theorem of Algebra

Study Guide

Key Definition

The Fundamental Theorem of Algebra states that every non-constant, single-variable polynomial of degree $n$ with complex coefficients has exactly $n$ complex roots, counting multiplicities.

Important Notes

A polynomial with real coefficients is a subset of polynomials with complex coefficients.
Every real number can be written as $a + bi$ where $b = 0$.
The theorem assures at least one complex root even if the polynomial seems not to have any.
Roots can be real or complex; real numbers are considered complex numbers with imaginary part zero.
Multiplicities must be counted to account for repeated roots.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

$\sqrt{x}$ denotes the square root of $x$

$x^n$ represents $x$ raised to the power of $n$

Remember to use proper notation when solving problems

Why It Works

The theorem ensures that every polynomial equation has solutions within the complex number system, thus providing a complete set of roots.

Remember

Every polynomial of degree $n$ has exactly $n$ roots in the complex plane, counting multiplicities.

Quick Reference

Polynomial Degree:Determines the number of roots: $n$ roots for degree $n$

Understanding Fundamental Theorem of Algebra

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Video explanation of this concept

Beginner

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Beginner Explanation

The Fundamental Theorem of Algebra states that any non-constant single-variable polynomial of degree $n$ with complex coefficients has exactly $n$ solutions in the complex number system, counting multiplicities. Repeated roots are each counted as many times as their multiplicity.

Practice Problems

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Quick Quiz

Single Choice Quiz

Beginner

What is the number of roots for a polynomial $x^3 + 2x^2 - 5$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Consider the quadratic polynomial $f(t) = -4.9t^2 + 24t + 1$, which models the height (in meters) of a basketball after it is released, as a function of time $t$ (in seconds). Describe what the roots of this polynomial represent in terms of the basketball’s trajectory.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given the polynomial $x^2 + 1$, find its complex roots.

Challenge Quiz

Single Choice Quiz

Advanced

Find the number of roots for $x^4 + x^3 - x + 1$.

Recap

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