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### Course: Algebra (all content) > Unit 10

Lesson 32: Zeros of polynomials and their graphs# Zeros of polynomials & their graphs

Learn about the relationship between the zeros, roots, and x-intercepts of polynomials. Learn about zeros multiplicities.

#### What you will learn in this lesson

When studying polynomials, you often hear the terms zeros, roots, factors and $x$ -intercepts.

In this article, we will explore these characteristics of polynomials and the special relationship that they have with each other.

## Fundamental connections for polynomial functions

For a polynomial $f$ and a real number $k$ , the following statements are equivalent:

is a$x={k}$ **root**, or solution, of the equation$f(x)=0$ is a${k}$ **zero**of function$f$ is an$({k},0)$ -intercept of the graph of$x$ $y=f(x)$ is a linear factor of$x-{k}$ $f(x)$

Let's understand this with the polynomial $g(x)=(x-3)(x+2)$ , which can be written as $g(x)=(x-3)(x-(-2))$ .

First, we see that the linear factors of $g(x)$ are $(x-{3})$ and $(x-({-2}))$ .

If we set $g(x)=0$ and solve for $x$ , we get $x={3}$ or $x={-2}$ . These are the solutions, or

**roots**, of the equation.A $x$ -value that makes the function value $0$ . Since we know $x=3$ and $x=-2$ are solutions to $g(x)=0$ , then ${3}$ and ${-2}$ are zeros of the function $g$ .

**zero**of a function is anFinally, the $x$ -intercepts of the graph of $y=g(x)$ satisfy the equation $0=g(x)$ , which was solved above. The $x$ -intercepts of the equation are $({3},0)$ and $({-2},0)$ .

### Check your understanding

## Zeros and multiplicity

When a linear factor occurs multiple times in the factorization of a polynomial, that gives the related zero

**multiplicity**.For example, in the polynomial $f(x)=(x-1)(x-4{)}^{{2}}$ , the number $4$ is a zero of ${2}$ .

**multiplicity**Notice that when we expand $f(x)$ , the factor $(x-4)$ is written ${2}$ times.

So in a sense, when you solve $f(x)=0$ , you will get $x=4$ twice.

In general, if $x-k$ occurs $m$ times in the factorization of a polynomial, then $k$ is a zero of multiplicity $m$ . A zero of multiplicity $2$ is called a

**double zero**.### Check your understanding

## The graphical connection

The multiplicity of a zero is important because it tells us how the graph of the polynomial will behave around the zero.

For example, notice that the graph of $f(x)=(x-1)(x-4{)}^{2}$ behaves differently around the zero $1$ than around the zero $4$ , which is a double zero.

Specifically, while the graphs $x$ -axis at $x=1$ , it only $x$ -axis at $x=4$ .

*crosses*the*touches*theLet's look at the graph of a function that has the same zeros, but different multiplicities. For example, consider $g(x)=(x-1{)}^{2}(x-4)$ . Notice that for this function $1$ is now a double zero, while $4$ is a single zero.

Now we see that the graph of $g$ $x$ -axis at $x=1$ and $x$ -axis at $x=4$ .

*touches*the*crosses*theIn general, if a function $f$ has a zero of $y=f(x)$ will $x$ -axis at that $x$ value. If a function $f$ has a zero of $y=f(x)$ will $x$ -axis at that point.

*odd*multiplicity, the graph of*cross*the*even*multiplicity, the graph of*touch*the### Check your understanding

### Challenge problem

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