WEBVTT
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Let's looks at polynomial functions
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of the variable x.
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Of course, the first thing we need to know
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is what is a polynomial function,
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and what is not.
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So, let's look at some examples.
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All of these are polynomial functions
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of the variable x.
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Notice that all of the x's are raised to a
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non-negative integer power.
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Square roots of numbers are okay, that's
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just a real number, but we want all the x's
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to be raised to a
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non-negative integer power.
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None of these are polynomials.
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The problem with the first one is there's an
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x in the denominator.
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The problem with the second one is there's
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an x under the square root, so that's not allowed.
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That's not a polynomial.
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This one has a negative exponent on the x,
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also not allowed, and this one has a
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fractional exponent on the variable x which
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is not allowed for a polynomial.
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For these that are polynomials that we can
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talk about the degree of the polynomial
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function of the variable x.
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What we would like to look at are the terms,
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and we want it multiplied out, not factored.
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So we want to pick out the exponent on the
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term that is the highest,
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the largest exponent.
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Alright so for this one, the highest
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exponent I see is 2 on a x,
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so the degree is 2.
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This one, the highest or largest exponent I
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see on x is a 1, so it has degree 1.
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And here I just want to be careful I'm not
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fooled: the largest exponent occurs
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on the last term.
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The largest exponent is 5,
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so this one has degree 5.
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Now let's see about
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other things about polynomials.
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We'd like to sketch a reasonable graph of a
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polynomial function, and there are several
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characteristics that will help us to do this.
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Let's look at a specific polynomial.
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Notice this one is factored
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very nicely for us.
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The first thing that I'd like to find is the
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y-intercept, and we know that, to find a
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y-intercept, we let x equal 0 and
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evaluate the function.
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So I want f of 0.
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That's going to be 2 times 0 squared times
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negative 4 times 2 cubed, and of course
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this is equal to 0.
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So, the first thing I know is I have a
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y-intercept at 0.
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Let's put that on our graph.
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The second thing I'd like to talk about are
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the x-intercepts and the behavior the graph
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does at each one.
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I'm going to make a chart to keep track of
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all the important information.
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Here I want to list my x-intercepts.
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In the center I want to
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list the multiplicity.
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The multiplicity of an x-intercept is the
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exponent on the factor from which you get
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that x-intercept.
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So, we'll talk about that more
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in just a moment.
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The last column is going to be
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for the behavior.
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Okay, so the x-intercepts are the values
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that would make this equal to 0, values of
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x, so certainly 0, positive 4, and negative 2.
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Okay, so the 0 comes from the factor x
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squared which has an exponent of 2, so the
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multiplicity of zero is 2.
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The 4 comes from the factor x minus 4
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which is to the first power,
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so the multiplicity is 1.
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And the negative 2 comes from x plus 2 cubed, the
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multiplicity must be 3.
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Now the behavior is either the graph
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will touch or cross the x-axis,
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and that depends on the multiplicity.
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Where the multiplicity is an even number,
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the graph will touch the x-axis without
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crossing, and where the multiplicity is an
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odd number, then the graph will cross
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through the x-axis.
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Let's put these x-intercepts on our graph.
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The x-intercept at 0, we already have that,
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the x-intercept at 4, and then the last
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x-intercept is at negative 2.
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Okay, the next thing we need to find out is
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what happens on the ends of the graph.
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So what is the power function the graph
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resembles for large absolute value of x?
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So we need to figure out, if we multiplied
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out the polynomial function, what would the
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term with the highest exponent look like?
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It would certainly have a factor of two,
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it would have an x,
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well we have this x squared.
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Now let's see what else.
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So from this term, we're going to be
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multiplying the 2 x squared times a positive
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x to the first, and then, from this one, the
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highest exponent I can get would be
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positive x to the third.
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And when we multiply like bases, we add the
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exponents so the power function the graph
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resembles for large absolute value of x is
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2 x to the sixth.
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Okay, 2 x to the sixth, this resembles
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positive x squared on the ends since the
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exponent is even and this coefficient is
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positive, so I can imagine the ends of the
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graph look like this, they both point up.
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Okay, one last question before we finish our
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graph: the maximum number of turning points
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for this function.
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This is always the degree minus one.
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Now we know the degree because it also
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happens to be the same exponent that's used
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on our power function.
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So, 6 minus 1 is 5.
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I should point out that the graph may not
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have five turning points, but it won't have
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more than five.
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Okay, so how can we get a graph?
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Well, the end behavior tells me that on the
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left of the leftmost x-intercept, the
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function values are doing
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something like this.
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In my chart I know that when I get to negative 2,
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the graph should cross the x-axis,
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and it comes down.
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Now, I really don't know how far down it
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goes, I would really need calculus
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to figure that out.
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But I do know at some point it comes down
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here and it turns and it comes back up to my
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next x-intercept.
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That one is at 0.
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The graph should touch at 0 without
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crossing, and then it comes back down.
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And again, I don't know how far down because
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I'd need calculus, but I do know eventually
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it has to turn because it has to hit the
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next x-intercept, which is 4,
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and the graph should cross there.
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Notice that the graph I've drawn is a smooth
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continuous curve, no sharp corners, no gaps,
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just a smooth continuous curve.
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Now I have one last question to answer.
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By looking at the graph, I can determine the
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intervals of x where the function f is above
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the x-axis and those on
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which is below the x-axis.
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So, if we start on the left, the graph is
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clearly above the x-axis from negative
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infinity to x equals negative 2, actually just close
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to negative 2, so we'll use parentheses on this to
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show that the negative 2 is not included.
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Right between negative 2 and 0, the graph is below
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the x-axis, so I'll list that here.
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Now the graph is still below the x-axis
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between 0 and 4, so we'll add that down
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here, it's below the x-axis.
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And then from 4 to infinity, the graph is
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above the x-axis again.
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Notice that this is two separate intervals:
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that's because at zero the graph touches, is
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on, the x-axis, so I should not include it.
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Let's look at another example.
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In this case, we're going to be given the
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graph of a polynomial function, and we need
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to choose a reasonable polynomial function
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from the list, something that
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might have this graph.
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Okay, so what kinds of things
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can we look at?
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We can look at the end behavior, and we can
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look at the x-intercepts and whether the
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graph touches or crosses at each one.
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So, let's start with this one.
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This x-intercept is at negative 2, and I see that
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the graph of the polynomial
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crosses the x-axis there.
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So, I need to find, in my possible answer, a
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factor of x plus 2 to an odd power.
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So, right away I can eliminate choice a
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because that's an even power, and I can
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eliminate choice b because the x plus 2 is
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to an even power again.
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However, this is x plus 2 to an odd power,
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so maybe this works, but so is this one
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since it's to the first, and here the x plus
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2 is cubed again, so any of these
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are still possible.
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Let's move on to another x-intercept.
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I have an x-intercept here at 0 and I notice
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that the graph touches the x-axis without
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crossing, and that means I need to find
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an x to an even power.
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Alright, well if we look at the three
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possibilities that are left, they're all the
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x's to an even power so I haven't
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eliminated another one.
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Move on to the other x-intercept at 1, and
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again the graph touches without crossing, so
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I need to see a factor of x minus 1 that's
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raised to an even power.
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And I have that here, I have that here, this
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isn't x minus 1 to an even power, but it
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still gives me an x-intercept at 1 with an
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even exponent so it's still
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going to touch there.
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So, I haven't eliminated anything else.
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The next thing I want to look at would be
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the end behavior.
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Okay, so on the left, I see the end behavior
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is something like this, and on the right,
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it looks like this.
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I want to try to think about what power
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function would have this sort of behavior.
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Well the power function is definitely going
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to have an odd exponent, so maybe x cubed,
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we can think about that.
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And that's close but x cubed actually looks
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like this, so what I have has been reflected
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across the x-axis, which means that it's
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going to be a negative x to some odd power.
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Okay, so I'll see what we have.
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For choice c, I want to look for the term
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with the highest exponent, so I need to
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imagine what I'd get if I can
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multiply it all out.
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So for c, I'd have 3 x to the fourth, here
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the highest power of x I could get
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would be x cubed.
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Notice it's a positive x there, this is a
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positive x and it would be squared, that
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would be the highest exponent.
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So that's 3 x to the ninth.
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It's got the odd exponent but not
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the negative coefficient.
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Choice c has been eliminated.
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Okay, now let's look at d.
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I have a negative x squared
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to start out with.
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And if I were to multiply this out, I would
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multiply it by the x there, and over here
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this is a positive x to the second power.
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Well let's see, that's negative x to the fifth.
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That one is good, that's one possibility.
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And then notice that more than one answer
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might be possible, so I don't want
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to give up yet.
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I need to look at letter e.
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Okay, so negative 5 x squared, if I multiplied this
000:11:40.545 --> 000:11:44.041
out, this would give me a x cubed.
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And here's where I have to be careful,
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the highest exponent would be the negative x squared.
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Alright, so overall this is
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negative 5 x to the seventh.
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Choice e is also an acceptable answer.