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## Integrated math 3

### Course: Integrated math 3>Unit 4

Lesson 2: Positive and negative intervals of polynomials

# Positive and negative intervals of polynomials

If we know all the zeros of a polynomial, then we can determine the intervals over which the polynomial is positive and negative. This is because the polynomial has the same sign between consecutive zeros. So all we need to do is check each interval that is between two consecutive zeros (or before the smallest zero and after the largest zero).

## Want to join the conversation?

• Think I've got this figured out:

Consecutive Negative Intervals
...if the zero is also a local maximum of the graph, then the graph will go up, with the highest point of that curve just touching the y-axis before going back down. When this happens, there will be two negative intervals in a row: (-)(-).

Consecutive Positive Intervals
...if the zero is also a local minimum of the graph, then the graph will drop down, with the lowest part of that curve just touching the y-axis before rising up again. When this happens, there will be two positive intervals in a row: (+)(+).

Alternating Intervals
...if the zero occurs where the graph completely crosses the y-axis, then you will have alternating intervals:
- if the graph is ascending: (-)(+)
- if the graph is descending: (+)(-)

Does this seem right?
• yes. there is an easy way of finding it; it is called the table of signs.
here is a link to kind of explain it.
https://youtu.be/idBbsanc8BQ
• How would you be able calculate at how high or low the points would go (or turning points of the graph) mathematically?
• Other than figuring out each point in between, this is a big part in calculus once you get to it.
One option though is to take the point exactly in between two consecutive zeroes. This will be a local max or min, so you would know that this "arc" of the graph will be symmetric around that max/ min (extrema). I hope that helps answering your question
• Did anyone else notice the sirens at ?
• this is SOOO hard to understand! can someone help? thanks! :D
• I'm pretty sure what Sal is saying is that the intervals between roots (zeroes) can be positive or negative, and you can find those roots by plugging in x values between your root values. Intervals will alternate between roots. Hope that helped.
• how did he get the samples did he just randomly pick them?
• Kind of. He had a set of numbers to pick from. So for instance the first was less than -2, so he just needed to pick a number less than -2, so he picked -3. He could have picked any number less than -2 though.

Some numbers are nicer than others when picking, For instance whole numbers are usually nicer than non whole rational numbers, and rational numbers are better than irrational number. 0 in particularly is a good choice when available, which Sal used when he could.

It kinda takes some practice to pick the easiest choices, but sticking to whole numbers or at least rational numbers should make it doable.
• he doesn't need to do all that math. once finds out that one is positive, its just positive, negative, positive, negative, etc.
• The polynomial y=x² is positive, then 0, then positive again. The sign of the polynomial doesn't always change when it has a zero.
• this doesn't make the practice section make sense. How can I know what are consecutive zeros and what aren't? If anyone helps me out, explain it like I'm 5 please lol
• Do the signs always have to alternate over intervals? For example, can we have a function that has zeros at a and b, then for the interval x < a it is positive, for a < x < b it is positive and continues to be positive for x > b? (i.e. an upward-opening parabola but forms somewhat of a semicircle where the minimum point typically is at)
• To have a continually positive or negative quadratic would be impossible. When you have a zero, the polynomial must cross the x-axis.
Looking at the interval when x < a and a < x < b as positive — which is possible — the polynomial must go down to hit b; thus, making the interval negative.

An excellent example of this:
f(x) = -(x+2)²(x+1)
Which when graphed on desmos visually shows the explanation.
(1 vote)
• Can someone explain "consecutive zeros"
(1 vote)
• A given polynomial will have a set of zeroes, which are inputs that make the polynomial equal 0. If we mark all of the real zeroes of a polynomial on the number line, then two zeroes are consecutive if there are no other zeroes between them.

For example, the polynomial x³-x has zeroes at -1, 0, and 1. -1 and 0 are consecutive, and 0 and 1 are consecutive, but -1 and 1 are not, since 0 is between them.
• At , shouldn't that be a positive 9 instead of negative?
At: , again shouldn't the answer be positive?
(1 vote)
• 2x-3
2(-3)-3
-6-3
-9

yes, you would get -9 from this