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question, and I'm not gonna write it the whole
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question. But it says to solve a polynomial inequality
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, we fact the polynomial into irreducible factors and find
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all the real. And there's a blank. And
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what we're trying to do is find the real roots
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. So the first blank that we're gonna fill in
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his roots. They were trying to find the real
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root to the polynomial. And then it says,
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Then we find the interval to turn by the rial
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roots sign of the polynomial on that interval. So
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again, this is the same blank bling and roots
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again, and then it gives us an example.
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It's his p of X is X X plus two
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and then x minus one, and it wants us
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to filling the diagram. So what they're doing is
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making a number line like this, and the roots
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are going to be minus two, which comes from
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this route from this factors Ari and then zero,
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which comes from this one. And then one is
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the last route, and it comes from this guy
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. So minus one plus one is zero. So
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we basically he opted science of in it. It
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actually asks what's the sign off a bunch of these
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X packs, plus two and then X minus one
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. And then finally, what's the sign of the
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whole thing together then? So, essentially, what
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we do is we look at a number that's less
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than minus two. Okay, so let's say minus
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three. So Axe is gonna be negative. Minus
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three plus two is also negative. Negative one minus
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three minus one is minus four. So it's negative
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. So there's three. Hang this over him all
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by three negatives. Together, we get a negative
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. Then I need a number between minus two and
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zero. So you kind of think of what I
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did is I plugged in minus three. That let's
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plug in minus one. So minus one lived in
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there is negative minus one plus two becomes positive minus
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one minus. One is negative. So that becomes
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positive. And then between zero and one. Let's
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say I put in 1/2 Okay, so X would
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be ah half. So it's positive we don't need
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to write in the actual value. We just need
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to write in whether it's positive or negative. So
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we're gonna say that's positive. Ah, half plus
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two is also positive and 1/2 minus. One is
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negative. So we get this to be negative.
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And then finally we take another bigger than one.
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So let's say two. Then that would mean that
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X would be too. So it's positive. X
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plus two would be four. Also, positive X
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minus one would be to minus one, which is
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one. So that's positive, which means the whole
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thing is positive. Okay, so then it says
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we see then from this diagram that P of X
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is greater than or equal to zero when it's positive
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. So it's on the intervals between minus two and
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zero. So X belongs the interval from minus 2
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to 0, and then the other one is from
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two to infinity because that's where they're positive. This
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guy's positive here, and this guy's positive. So
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between minus two and zero on them from to infinity
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did that some of you that question