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### Course: Integrated math 3>Unit 4

Lesson 2: Positive and negative intervals of polynomials

# 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 $y=f\left(x\right)$.
For example, let's suppose $f\left(x\right)=\left(x+3\right)\left(x-1{\right)}^{2}$. Since the zeros of function $f$ are $-3$ and $1$, the graph of $y=f\left(x\right)$ will have $x$-intercepts at $\left(-3,0\right)$ and $\left(1,0\right)$.
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 $f\left(x\right)=\left(x+1\right)\left(x-1\right)\left(x-3\right)$.
From the graph, we see that $f\left(x\right)$ is always ...
• ...negative when $-\mathrm{\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 $g\left(x\right)=x\left(x+2{\right)}^{2}$.
From the graph, we see that $g\left(x\right)$ is always...
• ...negative when $-\mathrm{\infty }.
• ...negative when $-2.
• ...positive when $0.
Notice that $g\left(x\right)$ does not change sign around $x=-2$.

## Determining the positive and negative intervals of polynomials

Let's find the intervals for which the polynomial $f\left(x\right)=\left(x+3\right)\left(x-1{\right)}^{2}$ 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 $-\mathrm{\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 $f\left(-4\right)$.
Because we are only interested in the sign of the polynomial here, we don't have to completely evaluate it:
$\begin{array}{rl}f\left(x\right)& =\left(x+3\right)\left(x-1{\right)}^{2}\\ \\ f\left(-4\right)& =\left(-4+3\right)\left(-4-1{\right)}^{2}\\ \\ & =\left(-\right)\left(-{\right)}^{2}& & \text{Evaluate only the sign of the answer.}\\ \\ & =\left(-\right)\left(+\right)& & \text{A negative squared is a positive.}\\ \\ & =-& & \text{A negative times a positive is a negative.}\end{array}$
Here we see that $f\left(-4\right)$ is negative, and so $f\left(x\right)$ will always be negative for $-\mathrm{\infty }.
We can repeat the process for the remaining intervals.
The results are summarized in the table below.
IntervalThe value of a specific $f\left(x\right)$ within the intervalSign of $f$ on intervalConnection to graph of $f$
$-\mathrm{\infty }$f\left(-4\right)<0$negativeBelow the $x$-axis
$-3$f\left(0\right)>0$positiveAbove the $x$-axis
$1$f\left(2\right)>0$positiveAbove the $x$-axis
This is consistent with the graph of $y=f\left(x\right)$.

1) $g\left(x\right)=\left(x+1{\right)}^{2}\left(x+6\right)$ has zeros at $x=-6$ and $x=-1$.
What is the sign of $g$ on the interval $-6?

2) $h\left(x\right)=\left(3-x\right)\left(x+5\right)\left(x-2\right)$ has zeros at $x=-5$, $x=2$, and $x=3$.
What is the sign of $h\left(x\right)$ on the interval $-5?

### Challenge problem

3*) Which of the following could be the graph of $g\left(x\right)=\left(x-2{\right)}^{2}\left(x+1{\right)}^{3}$?

## 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
• 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)
• 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.
• 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.
• 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"?
• 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.