Zeros of polynomials & their graphs

CCSS Math: 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 $(\tealD{-2},0)$.

1) What are the zeros of $f(x)=(x+4)(x-7)$?

We know that if $x-\tealD k$ is a linear factor of $f(x)$, then $\tealD k$ is a zero of function $f$.
We can rewrite $f(x)=(x+4)(x-7)$ as $f(x)=(x-(\tealD{-4}))(x-\tealD7)$. Therefore, the zeros of $f$ are $\tealD{-4}$ and $\tealD 7$.
2) The graph of function $g$ crosses the $x$-axis at $(2,0)$. What must be a root of the equation $g(x)=0$?
$x=$

If $(\tealD k,0)$ is an $x$-intercept of the graph of a function $f$, then we know that $x=\tealD k$ is a root, or solution, of the equation $f(x)=0$.
Since the graph of $g$ crosses the $x$-axis at $(\tealD2,0)$, then $x=\tealD2$ is a root of $g(x)=0$.
3) The zeros of function $h$ are $-1$ and $3$. Which of the following could be $h(x)$?

We know that if $\tealD k$ is a zero of function $f$, then $x-\tealD k$ is a linear factor of $f(x)$.
Since the zeros of function $h$ are $-1$ and $3$, we know that $(x-(-1))$ and $(x-3)$ must be factors of $h$.
Simplifying, we see that the factors of $h(x)$ must be $(x+1)$ and $(x-3)$. The only option with these factors is: $h(x)=(x+1)(x-3)$

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.
$f(x)=(x-1)\purpleC{(x-4)(x-4)}$
So in a sense, when you solve $f(x)=0$, you will get $x=4$ twice.
\begin{aligned}0&=(x-1)\purpleC{(x-4)(x-4)}\\ \\ &x-1=0\qquad x-4=0\qquad x-4=0\\\\ &x=1\qquad \qquad \purpleC{x=4}\qquad \qquad \purpleC{x=4} \end{aligned}
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.

4) Which zero of $f(x)=(x-3)(x-1)^3$ has multiplicity $3$?

$f(x)=(x-3)(x-1)^\tealD{3}$
Notice that $(x-1)$ appears three times in the factorization of $f(x)$. The zero that relates to this factor is $1$. Therefore $1$ has a multiplicity of $3$.
5) Which zero of $g(x)=(x+1)^3(2x+1)^2$ is a double zero?

$g(x)=(x+1)^3(2x+1)^\tealD2$
Notice that $(2x+1)$ appears twice in the factorization of $g(x)$. The zero that relates to this factor is $-\dfrac12$.
\begin{aligned}2x+1&=0\\ \\ 2x&=-1\\ \\ x&=-\dfrac12 \end{aligned}
Therefore $-\dfrac12$ is a double zero.

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.

6) In the graphed function, is the multiplicity of the zero $6$ even or odd?

The graph of the polynomial crosses the $x$-axis at $(6,0)$. Therefore, $6$ is a zero with odd multiplicity.
7) Which is the graph of $h(x)=x^2(x-3)$?

The function has two zeros, $0$ and $3$, and so the graph of $y=h(x)$ must have $x$-intercepts at $(0,0)$ and $(3,0)$.
Because $x$ occurs twice in the factorization of $h$, the related zero, $0$, has a multiplicity of $2$. This is even, and so the graph of $h$ will touch the $x$-axis at $x=0$.
Because $x-3$ occurs once in the factorization of $h$, the related zero, $3$, has a multiplicity of $1$. This is odd, and so the graph of $h$ will cross the $x$-axis at $x=3$.
The only graph that reflects this is $B$.

Challenge problem

8*) Which is the graph of $f(x)=-x^3+4x^2-4x$?
\begin{aligned} f(x)&=-x^3+4x^2-4x \\ \\&= -x(x^2-4x+4) \\ \\ &= -x(x-2)(x-2)\\ \\&=-x(x-2)^2 \end{aligned}
From this we see that $0$ is a zero with odd multiplicity and $2$ is a zero with even multiplicity. So the graph of $f$ will cross the $x$-axis at $(0,0)$ and touch the $x$-axis at $(2,0)$.
The only option to reflect this is graph $D$