WEBVTT
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here. Let's go ahead and find the partial fraction
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to composition once we do so it's not necessary to
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find the values of the coefficient here. So here's
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our fraction. The first thing we should try to
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do is look at that denominator and see if the
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factors we always wanted found there as much as we
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can. Here it's important. And this the nominated
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It looks like we could take out an X squared
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and we're left over with one plus x squared and
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we'LL still see if we can factor. So looking
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at this first term here, X squared. This
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puts us in what the book calls case, too
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. We have a repeated linear factor in this case
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. It happens to be X. So there's X
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and it appears twice second power. And then we'LL
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have to also look at this for dramatic here.
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We'd like to know if that factors that this might
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factor into too linear polynomial sze. So given a
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polynomial a X square plus B X plus C.
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If you'd like to know whether you can factor this
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thing using real numbers, this does not matter if
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B squared minus four A. C is less than
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zero. And our problem here the A, which
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is the constant in front of the X Square,
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is just one be a zero. There's no ex
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term, so be has to be zero. And
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we see the constant term C is also one and
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therefore B squared minus four. A c equals negative
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for this's negative number. So x squared plus one
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does not back there. And so this is what
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the book calls case story. This is when you
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have a quadratic factor that doesn't factor into too linear
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polynomial sze or what some might call an irreducible quadratic
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. So now putting case to in cases three together
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. So looking at the first term, the one
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over X squared Listen, be color coordinated here we
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have in Blue Rex because X is the linear factor
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. And then since X goes to the second power
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we do another term and we go all the way
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up until we hit the highest power of X and
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it just happens to be two. So we'LL stop
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there for the one over X squared And now for
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the quadratic using history, we'LL have that quadratic back
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on the bottom on the denominator, But then on
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the numerator, instead of having a constancy, we
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need to have a linear polynomial appear. So this
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is case three. So we'LL put a CX plus
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de So this is it could be any polynomial that's
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linear for part me. The first thing we should
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do here I actually have to go to the next
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page for part B because the numerator is degree three
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denominators degree three, We should do polynomial division.
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So let's go to the next page and do long
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division here. So let's rewrite that polynomial or this
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case the rational function X cubed minus one Execute minus
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three X square plus two x Let's go ahead and
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divide this What? So x cued times one is
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execute so well guarded multiplied this all by one and
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then we subtract and we're left over with three X
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square minus two works minus one and we can no
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longer divide because this is a larger power of X
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. Then we have down here. So this tells
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us that the original fraction could be written as one
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, and then we're left over with the remainder,
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and then we have the original denominator and now we
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can consider the partial fractions because now the numerator has
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degree That's smaller than the denominator. So now we
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tried a factor. The denominator as much as we
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can. And we could do the partial fraction the
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composition. So in the denominator, the first thing
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we could do is pull out of next term so
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that we have a quadratic left over and this quadratic
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wolf after in the denominator we have X and then
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we have X minus two X minus one. And
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now this is what the book calls Case one.
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We have linear factors that are all this thing.
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So we'LL have one linear factor. They get a
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constant on top. So we'll have this thing Constance
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A, B and C for each one of these
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terms and that's your answer for part B.