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Polynomials with Complex Roots

Learn how to find and interpret complex roots of quadratic polynomials.

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Polynomials with Complex Roots

Study Guide

Key Definition

The Fundamental Theorem of Algebra states that any polynomial with real coefficients can be factored completely over the field of complex numbers.

Important Notes

The roots are complex when the discriminant is negative.
Complex roots come in conjugate pairs like $-5 + 12i$ and $-5 - 12i$ (for polynomial $x² + 10x + 169$).
Factor quadratic polynomials using the quadratic formula.
Use $i$ to represent $√(-1)$.
Complex numbers are in the form $a + bi$.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$i$ represents the imaginary unit

$√a$ represents the square root of $a$

$x²$ represents $x$ squared

Remember to use proper notation when solving problems

Why It Works

Complex roots allow polynomials to be factored completely, ensuring all solutions are found.

Remember

Use the quadratic formula $\frac{-b \pm √(b² - 4ac)}{2a}$ to find roots.

Quick Reference

Quadratic Formula:$\frac{-b \pm √(b² - 4ac)}{2a}$

Imaginary Unit:$i = √(-1)$

Understanding Polynomials with Complex Roots

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A quadratic equation has complex roots when its discriminant (b²–4ac) is negative. We use x = (–b ± √(b²–4ac))/(2a), writing √(negative number) as i·√(positive), to get two conjugate roots.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What are the complex roots of $x² + 10x + 169$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you're designing a roller coaster loop whose shape is modeled by the equation $x² + 10x + 169 = 0$. A negative discriminant means the loop is closed with no real intercepts.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Determine the nature of roots for $x² + 4x + 5$.

Challenge Quiz

Single Choice Quiz

Advanced

Find the roots of $2x² + 4x + 8$.

Recap

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