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So we're gonna win this inequality X cubed plus X
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squared minus 17 X plus 15 and were asked to
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find out when that's greater than or equal to zero
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. So the first thing we have to do is
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find the roots. So to do that, we
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have the factor. This And in order to do
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that, we have to use synthetic substitution and recognize
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that the possible factors are one plus or minus.
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Uh, plus or minus three. Let's remind us
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five and plus or minus 15. I'm gonna try
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one, and I can see that one works because
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one cubed plus one squared minus 17 times one plus
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15 gives me zero. So I know that one
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is a route. So then I'm gonna divide by
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synthetic substitution. So I'm gonna go one and then
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11 minus 17 and 15 and I'm gonna bring the
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one down first and then one times one is one
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. If I had to get to one. Times
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two is two minus 17 plus two is minus 15
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times one is minus 15 and I get zero.
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So this thing factors now as X minus one,
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because one is the root on two X squared plus
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two x minus 15 is created that are equal to
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zero. And then that factors as X minus one
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times X close. Five X minus three is greater
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than or equal to zero. So that leaves me
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in terms of my number line with three possible routes
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, minus five one and then three. And all
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I do is I take a number that's less than
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that. So let's say four. Sorry not for
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, let's say, minus six. First, we're
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going there and I plug it into each of these
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. So minus six minus one is negative. Minus
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six plus five is negative and might have six.
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Minus three is negative. So I get three negatives
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, which makes this negative this interval. Negative.
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Then I plug in, Let's say, minus four
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in here, so minus four minus one is minus
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five. So it's negative. This is positive,
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though, cause it's minus four plus five and then
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minus four minus three is negative. So I get
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a positive here and then between one and three.
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I could put in two and I will get a
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negative and putting something bigger than three. Let's say
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four, I'll get a positive, and I'm confident
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that that is the case because these are all single
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roots. And if you have gone through the polynomial
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rational function chapter, the start of this chapter,
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you know that if it's a single room, it
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means it has to change. Sign when we look
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at the positive intervals are solution is going to be
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the interval from minus five upto one. It's positive
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and then from three to infinity. So that is
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our solution, so it's a bit of involved mathematics
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.