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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=f(x).
For example, let's suppose f(x)=(x+3)(x1)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

An x y coordinate plane with a point at (negative three, zero) and a point at (one, zero).
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.
A polynomial labeled y equals f of x is graphed on an x y coordinate plane. The graph curves up from left to right passing through the x-axis at negative one, zero. It curves down and passes through the x-axis at one, zero. It curves back up and passes through the x-axis at three, zero. From negative infinity to negative one on the x-axis, the section below the x-axis is shaded and labeled negative. From negative one to one on the x-axis is shaded above the x-axis and labeled positive. From one to three on the x-axis is shaded below the x-axis and labeled negative. From three to positive infinity on the x-axis the region is shaded above the x-axis and labeled positive.
For example, consider the graphed function f(x)=(x+1)(x1)(x3).
From the graph, we see that f(x) is always ...
  • ...negative when <x<1.
  • ...positive when 1<x<1.
  • ...negative when 1<x<3.
  • ...positive when 3<x<.
It is not necessary, however, for a polynomial function to change signs between zeros.
A polynomial labeled y equals g of x is graphed on an x y coordinate plane. The graph curves up from left to right touching the x-axis at (negative two, zero) curving down. It curves back up and passes through the x-axis at (zero, zero). From negative infinity to negative two on the x-axis, the section below the x-axis is shaded and labeled negative. From negative two to zero on the x-axis is shaded below the x-axis and labeled negative. From zero to infinity on the x-axis is shaded above the x-axis and labeled positive.
For example, consider the graphed function g(x)=x(x+2)2.
From the graph, we see that g(x) is always...
  • ...negative when <x<2.
  • ...negative when 2<x<0.
  • ...positive when 0<x<.
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)(x1)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:
An x y coordinate plane with a point at (negative three, zero) and a point at (one, zero). A vertical dashed line passes through the point at (negative three, zero). A vertical dashed line passes through the point at (one, zero).
Let’s find the sign of f for <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:
f(x)=(x+3)(x1)2f(4)=(4+3)(41)2=()()2Evaluate only the sign of the answer.=()(+)A negative squared is a positive.=A negative times a positive is a negative.
Here we see that f(4) is negative, and so f(x) will always be negative for <x<3.
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
<x<3f(4)<0negativeBelow the x-axis
3<x<1f(0)>0positiveAbove the x-axis
1<x<f(2)>0positiveAbove the x-axis
This is consistent with the graph of y=f(x).
An x y coordinate plane with a point at (negative three, zero) and a point at (one, zero). A vertical dashed line passes through the point at (negative three, zero). A vertical dashed line passes through the point at (one, zero). From negative infinity to negative three on the x-axis, the section below the x-axis is shaded and labeled negative. From negative three to one on the x-axis is shaded above the x-axis and labeled positive. From one to infinity on the x-axis is shaded above the x-axis and labeled positive.

Check your understanding

1) g(x)=(x+1)2(x+6) has zeros at x=6 and x=1.
What is the sign of g on the interval 6<x<1?
Choose 1 answer:

2) h(x)=(3x)(x+5)(x2) has zeros at x=5, x=2, and x=3.
What is the sign of h(x) on the interval 5<x<2?
Choose 1 answer:

Challenge problem

3*) Which of the following could be the graph of g(x)=(x2)2(x+1)3?
Choose 1 answer:

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?

  • leafers ultimate style avatar for user Staci
    Is there a section that covers finding the zeros of polynomials using p/q and synthetic division?
    (19 votes)
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  • starky ultimate style avatar for user Felicia L.
    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.
    (18 votes)
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    • primosaur ultimate style avatar for user Jordan Cooper
      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.
      (9 votes)
  • blobby green style avatar for user Sruja Konada
    im still confused as to how to find if the interval is positive or negative
    (19 votes)
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    • duskpin tree style avatar for user WolfyNightshade
      Positive interval: The function is above the x-axis
      Negative interval: The function is below the x-axis

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

      If you don't have a graph, then you need to test 1 or 2 values within the function.
      If Y is positive, then the interval is positive.
      If Y is negative, then the interval is negative.
      (1 vote)
  • blobby green style avatar for user Dakota 9000
    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.
    (10 votes)
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    • stelly blue style avatar for user Kim Seidel
      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.
      (6 votes)
  • aqualine ultimate style avatar for user Aunnesha
    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)
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    • stelly blue style avatar for user Kim Seidel
      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.
      (13 votes)
  • blobby green style avatar for user oliverhendleman
    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
    (3 votes)
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    • female robot grace style avatar for user loumast17
      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.
      (8 votes)
  • primosaur ultimate style avatar for user Tomatomath
    i'm also confused by (no breaks in the graph)as the meaning of
    "continuous funtion"

    can anybody explaine to me more detail by "no breaks in the graph"?
    (4 votes)
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    • stelly blue style avatar for user Kim Seidel
      It is exactly what it says. You will have a line/curve that extends from the left side to the right side of the coordinate plane, and there will be no gaps. Or, you can think of the equation / function as being defined for inputs values (x) = all real numbers. No values are excluded. An exclusion would leave a gap.
      (4 votes)
  • blobby green style avatar for user keshav rawat
    How do we figure out the maximum or minimum point the graph of polynomial will take?
    (3 votes)
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  • hopper jumping style avatar for user Sam D
    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?
    (3 votes)
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    • leaf green style avatar for user Mikey Schwartz
      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.
      (4 votes)
  • primosaur ultimate style avatar for user Tomatomath
    SO, is there any example that the sign of a polynomil between any two concecutive zero is "BOTH NEGATIVE AND POSSITIVE"?
    (3 votes)
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    • leaf green style avatar for user kubleeka
      No, because of the 'consecutive' condition, meaning there are no zeros between the two given zeros. If the polynomial has zeros at a and b, and switches between positive and negative in the interval [a, b], then there must be another zero between a and b where the sign change occurred.
      (3 votes)