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Roots

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Roots

Study Guide

Key Definition

A root is a value of a variable that makes a function equal zero. For example, in the polynomial $f(x) = x^2 - 4$, the roots are $x = -2$ and $x = 2$.

Important Notes

A degree-n polynomial has n roots over the complex numbers (counting multiplicities), though it may have fewer real roots.
Roots can be real or complex.
Multiple roots are possible.
The fundamental theorem of algebra guarantees roots for non-zero polynomials.
Practicing root finding strengthens algebra skills.

Mathematical Notation

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

$x^2$ represents $x$ squared

$+$ represents addition

$-$ represents subtraction

$=$ represents equality

Remember to use proper notation when solving problems

Why It Works

Finding roots involves solving equations where the function equals zero. This is fundamental to understanding polynomial behavior.

Remember

The roots of a polynomial are solutions to the equation $f(x) = 0$.

Quick Reference

Square Root:$\sqrt{x}$

Quadratic Formula:$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Degree of a Polynomial:$n$ (degree of the polynomial; exactly n complex roots counting multiplicities, possibly fewer real roots)

Understanding Roots

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Roots are the values that make a polynomial equal zero. For example, in $x^2 - 4$, the roots are $2$ and $-2$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What are the roots of $x^2 - 4$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you're designing a square garden plot where the side length x satisfies $x^2 - 6x + 9 = 0$. Find the roots.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Find the roots of $2x^2 - 8x + 6 = 0$ using the quadratic formula.

Challenge Quiz

Single Choice Quiz

Advanced

What are the roots of $x^3 - 3x^2 + 3x - 1$?

Recap

Watch & Learn

Review key concepts and takeaways