1.1: Zeros and Multiplicity | Precalculus

How it works

A factor x - r produces a zero at r. Its exponent is the zero multiplicity. Odd multiplicity crosses the x-axis; even multiplicity touches and turns. Multiplicity helps connect factored equations to a graph without expanding the polynomial.

f(x)=(x-r)^m q(x),\ q(r)\ne0

Worked examples

Example 1

Analyze the zeros of $\displaystyle f(x)=(x+2)^2(x-1)$ .

$\displaystyle x=-2$ has multiplicity $\displaystyle 2$ and touches the axis. $\displaystyle x=1$ has multiplicity $\displaystyle 1$ and crosses.

Example 2

Write a monic cubic with a double zero at $\displaystyle 3$ and a single zero at $\displaystyle -1$ .

The required factors are $\displaystyle (x-3)^2$ and $\displaystyle (x+1)$ , so $\displaystyle f(x)=(x-3)^2(x+1)$ .

Problems

Problem 1.1.1

List zeros and multiplicities of $\displaystyle (x-4)^3(x+1)^2$ .

Show worked solution

$\displaystyle 4$ has multiplicity $\displaystyle 3$ and $\displaystyle -1$ has multiplicity $\displaystyle 2$ . The graph crosses at $\displaystyle 4$ and touches at $\displaystyle -1$ .

Problem 1.1.2

Write a monic quadratic with zeros $\displaystyle -2$ and $\displaystyle 5$ .

Show worked solution

$\displaystyle f(x)=(x+2)(x-5)=x^2-3x-10$ .

Problem 1.1.3

Use the graph of $\displaystyle y=(x+1)^2(x-2)$ to describe its zeros and whether it crosses or touches.

The cubic y = (x + 1) squared times (x − 2) touches the axis at −1 and crosses at 2. — Compare the graph at its two zeros.

Show worked solution

The zero $\displaystyle -1$ has multiplicity $\displaystyle 2$ and touches the axis. The zero $\displaystyle 2$ has multiplicity $\displaystyle 1$ and crosses.

The cubic y = (x + 1) squared times (x − 2) touches the axis at −1 and crosses at 2. −1 has multiplicity 2; 2 has multiplicity 1. — −1 has multiplicity 2; 2 has multiplicity 1.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 1.1.1

Problem 1.1.2

Problem 1.1.3

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