Polynomials With Complex Roots - Varsity Tutors

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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 $x$ squaredRemember 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

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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.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

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

A$-5 + 12i$ and $-5 - 12i$B$5 + 12i$ and $5 - 12i$C$-10 + 13i$ and $-10 - 13i$D$10 + 12i$ and $10 - 12i$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

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.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

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

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

Find the roots of $2x² + 4x + 8$.

A$-1 + i√3$ and $-1 - i√3$B$1 + i√3$ and $1 - i√3$C$-2 + i√2$ and $-2 - i√2$D$2 + i√2$ and $2 - i√2$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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