Master

Repeated Zeros

Master repeated zeros with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

Repeated Zeros

Study Guide

Key Definition

Repeated zeros of a polynomial function occur when a factor appears more than once in the factored form of the polynomial, such as $(x - 3)^2$.

Important Notes

A repeated zero is also known as a root with multiplicity greater than one.
If $(x - a)^2$ is a factor, $x = a$ is a repeated zero.
Repeated zeros affect the shape of the graph at the zero point.
A double root will cause the graph to touch and bounce off the x-axis.
The multiplicity of a zero affects the derivative of the polynomial.

Mathematical Notation

$(x - a)^n$ indicates a zero at x = a with multiplicity n

$x = a$ denotes the location of a root

$n$ represents how many times the factor repeats

Use exponent notation to denote repeated zeros

Why It Works

Repeated zeros arise from repeated factors in polynomials, such as $(x - a)^n$, impacting the polynomial's graph and derivative.

Remember

Repeated zeros occur when factors are repeated in the polynomial's factorization, such as $(x - 3)^2$.

Quick Reference

Repeated Zero:$(x - a)^n$

Understanding Repeated Zeros

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A repeated zero occurs when a factor in the polynomial, such as $(x - 3)^2$, appears more than once.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the repeated zero of $f(x) = (x - 3)^2$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Find the repeated zero of $f(x) = (x - 2)^3$. A skateboard ramp is designed with a polynomial function.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Explore the impact of repeated zeros on the polynomial $f(x) = (x - 4)^2(x + 1)$.

Challenge Quiz

Single Choice Quiz

Advanced

Determine the nature of zeros for $f(x) = (x - 1)^3(x + 2)^2$.

Recap

Watch & Learn

Review key concepts and takeaways