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Real Zero of a Function

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Real Zero of a Function

Study Guide

Key Definition

A real zero of a function $f$ is a real number $r$ such that $f(r) = 0$.

Important Notes

The equation $f(x) = 0$ is used to find real zeros.
Real zeros are where the graph crosses the x-axis.
A polynomial of degree $n$ can have up to $n$ real zeros.
Factoring a polynomial can help find its zeros.
Complex zeros come in conjugate pairs, but they are not real zeros.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

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

Remember to use proper notation when solving problems

Why It Works

The real zero of a function $f(x)$ occurs where the function equals zero, which is equivalent to solving $f(x) = 0$.

Remember

To find real zeros, solve $f(x) = 0$.

Quick Reference

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

Understanding Real Zero of a Function

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Video explanation of this concept

Beginner

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Beginner Explanation

A real zero is where a function $f(x)$ crosses the x-axis, meaning $f(x) = 0$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What are the real zeros of $f(x) = x^2 - 3x + 2$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboard ramp follows a quadratic path. Find the x-coordinates where $y = 0$ for the ramp described by $y = x^2 - 2x - 8$.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider a cubic function $f(x) = x^3 - 6x^2 + 11x - 6$. Find its real zeros.

Challenge Quiz

Single Choice Quiz

Advanced

What are the zeros of $f(x) = x^3 - x^2 - x + 1$?

Recap

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Review key concepts and takeaways