# Zeros of polynomials & their graphs

CCSS Math: HSA.APR.B.3, HSF.IF.C.7, HSF.IF.C.7c

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:

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

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 $(x-\tealD3)$ and $(x-(\tealD{-2}))$.

If we set $g(x)=0$ and solve for $x$, we get $x=\tealD3$ or $x=\tealD{-2}$. These are the solutions, or

**roots**, of the equation.A

**zero**of a function is an $x$-value that makes the function value $0$. Since we know $x=3$ and $x={-2}$ are solutions to $g(x)=0$, then $\tealD3$ and $\tealD{-2}$ are zeros of the function $g$.Finally, 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 $(\tealD3,0)$ and $(\tealD{-2},0)$.

### 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 $f(x)=(x-1)(x-4)^\purpleC{2}$, the number $4$ is a zero of

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

*crosses*the $x$-axis at $x=1$, it only*touches*the $x$-axis at $x=4$.Let'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 $g$

*touches*the $x$-axis at $x=1$ and*crosses*the $x$-axis at $x=4$.In general, if a function $f$ has a zero of

*odd*multiplicity, the graph of $y=f(x)$ will*cross*the $x$-axis at that $x$ value. If a function $f$ has a zero of*even*multiplicity, the graph of $y=f(x)$ will*touch*the $x$-axis at that point.