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Positive & negative intervals of polynomials

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, equals, f, left parenthesis, x, right parenthesis.
For example, let's suppose f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 3, right parenthesis, left parenthesis, x, minus, 1, right parenthesis, squared. Since the zeros of function f are minus, 3 and 1, the graph of y, equals, f, left parenthesis, x, right parenthesis will have x-intercepts at left parenthesis, minus, 3, comma, 0, right parenthesis and left parenthesis, 1, comma, 0, right parenthesis.
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, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, minus, 1, right parenthesis, left parenthesis, x, minus, 3, right parenthesis.
From the graph, we see that f, left parenthesis, x, right parenthesis is always ...
• ...negative when minus, infinity, is less than, x, is less than, minus, 1.
• ...positive when minus, 1, is less than, x, is less than, 1.
• ...negative when 1, is less than, x, is less than, 3.
• ...positive when 3, is less than, x, is less than, infinity.
It is not necessary, however, for a polynomial function to change signs between zeros.
For example, consider the graphed function g, left parenthesis, x, right parenthesis, equals, x, left parenthesis, x, plus, 2, right parenthesis, squared.
From the graph, we see that g, left parenthesis, x, right parenthesis is always...
• ...negative when minus, infinity, is less than, x, is less than, minus, 2.
• ...negative when minus, 2, is less than, x, is less than, 0.
• ...positive when 0, is less than, x, is less than, infinity.
Notice that g, left parenthesis, x, right parenthesis does not change sign around x, equals, minus, 2.

Determining the positive and negative intervals of polynomials

Let's find the intervals for which the polynomial f, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 3, right parenthesis, left parenthesis, x, minus, 1, right parenthesis, squared is positive and the intervals for which it is negative.
The zeros of f are minus, 3 and 1. This creates three intervals over which the sign of f is constant:
Let’s find the sign of f for minus, infinity, is less than, x, is less than, minus, 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 minus, 4 is in this interval, let's find f, left parenthesis, minus, 4, right parenthesis.
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 &&{\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{aligned}
Here we see that f, left parenthesis, minus, 4, right parenthesis is negative, and so f, left parenthesis, x, right parenthesis will always be negative for minus, infinity, is less than, x, is less than, minus, 3.
We can repeat the process for the remaining intervals.
The results are summarized in the table below.
IntervalThe value of a specific f, left parenthesis, x, right parenthesis within the intervalSign of f on intervalConnection to graph of f
minus, infinity, is less than, x, is less than, minus, 3f, left parenthesis, minus, 4, right parenthesis, is less than, 0negativeBelow the x-axis
minus, 3, is less than, x, is less than, 1f, left parenthesis, 0, right parenthesis, is greater than, 0positiveAbove the x-axis
1, is less than, x, is less than, infinityf, left parenthesis, 2, right parenthesis, is greater than, 0positiveAbove the x-axis
This is consistent with the graph of y, equals, f, left parenthesis, x, right parenthesis.

1) g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, plus, 1, right parenthesis, squared, left parenthesis, x, plus, 6, right parenthesis has zeros at x, equals, minus, 6 and x, equals, minus, 1.
What is the sign of g on the interval minus, 6, is less than, x, is less than, minus, 1?

2) h, left parenthesis, x, right parenthesis, equals, left parenthesis, 3, minus, x, right parenthesis, left parenthesis, x, plus, 5, right parenthesis, left parenthesis, x, minus, 2, right parenthesis has zeros at x, equals, minus, 5, x, equals, 2, and x, equals, 3.
What is the sign of h, left parenthesis, x, right parenthesis on the interval minus, 5, is less than, x, is less than, 2?

Challenge problem

3*) Which of the following could be the graph of g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 2, right parenthesis, squared, left parenthesis, x, plus, 1, right parenthesis, cubed?

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.

Want to join the conversation?

• Is there a section that covers finding the zeros of polynomials using p/q and synthetic division?
• Once you know whether the first interval is positive or negative, isn't it easy to tell what the rest of the intervals are based on the multiplicity of the zeros? The graph will always either touch or cross the x axis at every zero.
• Correct.

You could start with any interval and work away from it. If an interval includes zero, I find that easiest to check in factored form. Or if you also have the unfactored polynomial, the intervals on the ends might be obvious based on the leading term.

But the methods in this article work even if you haven't learned about the behavior of multiple roots yet.
• im still confused as to how to find if the interval is positive or negative
• all of this is total shenanigans in the first skill for positive and negative intervals for polynomials it says that an interval can be both positive and negative when it is clearly only one. In the hints, it even looks like they changed the interval to fit their answer. Please explain I've already watched the video for this and the article spending many hours trying to figure this out.
• Positive interval: The points for the function, or the graph sits above the x-axis
Negative interval: The points for the function, or the graph sits below the x-axis

If you have a graph, this is very easy - look at the graph and see if the line for the function sits above or below the x-axis.

If you don't have a graph, then you need to test 1 or 2 values.
-- If Y turns out to be positive, then the interval is positive because the point sits above the line.
-- If Y turns out to be negative, then the interval is negative because the point sits below the line.

For example, for the 1st problem to "Check Your Understanding", I used a value of x=-2 which is in the given interval. Plug in -2 for x in the function g(x)=(x+1)^2(x+6)
g(-2) = (-2+1)^2(-2+6) = (-1)^2(4) = 1(4) = +4
The interval is positive because the point (-2, 4) is above the x-axis.

Hope this helps.
• When doing problems with positive and negative intervals of polynomials and it asks what the sign of f(x) is, given only the zeros (not the graph itself), how do we know if the graph starts from below the point or above the point? Because the graph could be drawn two ways right?
(1 vote)
• Read thru this article again and try the practice problems. Sal gives an example that has no graphs. You basically have to test values that fall between the x-intercepts. And, as Sal shows above, you don't even need to do the complete math. Once you pick the number you want to test, you can just see what sign that creates.
• How do we figure out the maximum or minimum point the graph of polynomial will take?
• To do this for an arbitrary polynomial, you need an understanding of calculus. I would suggest building up your trigonometry, geometry, and algebra background before diving into calculus!
• The articles says in positive and negative intervals Why? which means that the only way to change signs is to cross the x-axis. But if this happened, the given zeros would not be consecutive! I get that to change the sign the function needs to cross the x axis, but what does the latter part mean? Not consecutive anymore?
• What the article is saying is this: A zero of a function is where the function touches the x axis. In order for the sign of a polynomial to change (positive to negative or negative to positive on the y axis), the polynomial needs to cross the x axis. Wherever the polynomial crosses the x axis, there will be a zero. Therefore, if two zeros are consecutive, the function cannot cross the x axis between them (because then there would be a zero in between those two zeros, which isn't possible because they are consecutive), and if the function cannot cross the x axis, the sign of the function cannot change.
• Are you allowed to plug in 0 to the functions as a way to test their orientation as either positive/negative?
• Yes. Zero is as valid a number as any other when used to test for orientation.
• Could you explain how to solve this?

The polynomial p(x) = x^3 + 2x −11 has a real zero between which two consecutive integers?

(A) 0 and 1
(B) 1 and 2
(C) 2 and 3
(D) 3 and 4
(E) 4 and 5
• You could graph it and look at where the zeroes look like they are. visually you could test each of the options.

the alternative is to factor it and find the zeroes in factored form. This one really doesn't look like it factors nicely, so I think it wants you to just graph it.

If you don't have a graphing calculator you can find something online. Desmos is really good.

After you graph it check between each interval. look on the graph and see if the line crosses between 0 and 1, then 1 and 2 and so on.

Let me know if you still can't quite get it.