# Positive & negative intervals of polynomials

CCSS Math: HSA.APR.B.3, HSF.IF.C.7, 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 $-\infty<x<-1$.
- ...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.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:

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:

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.

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 $y=f(x)$.

### Check your understanding

### Challenge problem

### 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.