Positive & negative intervals of polynomials

CCSS Math: HSF.IF.C.7c
Learn about the relationship between the zeros of polynomials and the intervals over which they are positive or negative.

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

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.
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.
• ...positive when $-1.
• ...negative when $1.
• ...positive when $3.
It is not necessary, however, for a polynomial function to change signs between zeros.
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.
• ...negative when $-2.
• ...positive when $0.
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:
Let’s find the sign of $f$ for $-\infty.
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{aligned} f(x) &= (x+3)(x-1)^2\\ \\ f(-4) &= ({-4+3})({-4-1})^2 \\\\ &= ( -)(-)^2 &&\small{\gray{\text{Evaluate only the sign of the answer.}}}\\\\ &=(-)(+)&&\small{\gray{\text{A negative squared is a positive.}}}\\ \\ &=-&&\small{\gray{\text{A negative times a positive is a negative.}}}\end{aligned}
Here we see that $f(-4)$ is negative, and so $f(x)$ will always be negative for $-\infty.
We can repeat the process for the remaining intervals.
The results are summarized in the table below.
IntervalThe value of a specific $f(x)$ within the intervalSign of $f$ on intervalConnection to graph of $f$
$-\infty$f(-4)<0$negativeBelow the $x$-axis
$-3$f(0)>0$positiveAbove the $x$-axis
$1$f(2)>0$positiveAbove the $x$-axis
This is consistent with the graph of $y=f(x)$.