What you should be familiar with before taking this lesson

The zeros of a polynomial $f$ correspond to the $x$-intercepts of the graph of $y=f(x)$.

For example, let's suppose $f(x)=(x+3)(x-1)^2$. Since the zeros of function $f$ are $-3$ and $1$, the graph of $y=f(x)$ will have $x$-intercepts at $(-3,0)$ and $(1,0)$.

If this is new to you, we recommend that you check out our zeros of polynomials article.

What you will learn in this lesson

[[☃ image 1]]

While the $x$-intercepts are an important characteristic of the graph of a function, we need more in order to produce a good sketch.

Knowing the sign of a polynomial function between two zeros can help us fill in some of the gaps.

In this article, we'll learn how to determine the intervals over which a polynomial is positive or negative and connect this back to the graph.

Positive and negative intervals

The sign of a polynomial between any two consecutive zeros is either always positive or always negative. [[☃ explanation 1]]

[[☃ image 1]]

For example, consider the graphed function $f(x)=(x+1)(x-1)(x-3)$.

From the graph, we see that $f(x)$ is always ...

  • ...negative when $-\infty<x<-1$.
  • ...positive when $-1<x<1$.
  • ...negative when $1<x<3$.
  • ...positive when $3<x<\infty$.

It is not necessary, however, for a polynomial function to change signs between zeros.

[[☃ image 1]]

For example, consider the graphed function $g(x)=x(x+2)^2$.

From the graph, we see that $g(x)$ is always...

  • ...negative when $-\infty<x<-2$.
  • ...negative when $-2<x<0$.
  • ...positive when $0<x<\infty$.

Notice that $g(x)$ does not change sign around $x=-2$.

Determining the positive and negative intervals of polynomials

Let's find the intervals for which the polynomial $f(x)=(x+3)(x-1)^2$ is positive and the intervals for which it is negative.

The zeros of $f$ are $-3$ and $1$. This creates three intervals over which the sign of $f$ is constant:

[[☃ image 1]]

Let’s find the sign of $f$ for $-\infty<x<-3$.

We know that $f$ will either be always positive or always negative on this interval. We can determine which is the case by evaluating $f$ for one value in this interval. Since $-4$ is in this interval, let's find $f(-4)$.

Because we are only interested in the sign of the polynomial here, we don't have to completely evaluate it:

$\begin{align}
f(x) &= (x+3)(x-1)^2
\\
f(-4) &= ({-4+3})({-4-1})^2
\\
&= ( -)(-)^2 &&{\gray{\text{Evaluate only the sign of the answer.}}}
\\
&=(-)(+)&&{\gray{\text{A negative squared is a positive.}}}
\\
&=-&&{\gray{\text{A negative times a positive is a negative.}}}
\end{align}$

Here we see that $f(-4)$ is negative, and so $f(x)$ will always be negative for $-\infty<x<-3$.

We can repeat the process for the remaining intervals. [[☃ explanation 2]]

The results are summarized in the table below.

Interval | The value of a specific $f(x)$ within the interval| Sign of $f$ on interval|Connection to graph of $f$
:- | :-: | :-
$-\infty<x<-3$ | $f(-4)<0$ | negative| Below the $x$-axis
$-3<x<1$ | $f(0)>0$ | positive| Above the $x$-axis
$1<x<\infty$ | $f(2)>0$ | positive| Above the $x$-axis

This is consistent with the graph of $y=f(x)$.

[[☃ image 1]]

Check your understanding

[[☃ graded-group 1]]

[[☃ graded-group 1]]

Challenge problem

[[☃ graded-group 1]]

Determining positive & negative intervals from a sketch of the graph

Another way to determine the intervals over which a polynomial is positive or negative is to draw a sketch of its graph, based on the polynomial's end behavior and the multiplicities of its zeros.

Check out our graphs of polynomials article for further details.