BAM! mr. Tarrou in this precalculus video we're going to
be doing two examples or maybe algebra two honors of solving polynomial
inequalities using a graphical approach this is an addition to a lesson I've
done a number of years ago solving these polynomial inequalities using an
analytical approach I'll have a link to that older lesson in the description if
you would like to see a second technique we're gonna be solving this polynomial
inequality and graphing the solution set on a number line our first example we
have x cubed plus 2x squared minus 9x minus 18 is less than or equal to zero
in the start of this process is going to look like our other technique that's in
the other video which is we're going to solve this for 0 even if it wasn't part
of the inequality we would either be finding the critical values we're going
to use along the number line to test the interval set up the
intervals we will test or in this case using a graphical approach we want to
find those x-intercepts so anytime you have a polynomial with four terms or if
the degree is higher than to a higher order polynomial these can be quite a
bit of work to solve unless of course it's you can solve it through factoring
and that's what we're going to do here given those four terms quite often
you're going to be factoring by grouping so what common factor can we take out of
these first two terms well they both have they both have a factor of X their
coefficients don't have anything in common in terms of factors like 1 & 2
but we do have a common factor of X so we're going to take out that lowest
power so we're going to factor out that lowest power so we're going to factor
out that x squared when factoring out a monomial of course we do that through
division C inverse math operation multiplication which is what we do when
we distribute through parentheses X cubed divided by x squared well if you
have three and you took out two that just leaves you with one and we have two
x squared and divided by x squared of course the x squared is going to divide
out to be 1 leaving you with 2 times 1 which is
equal to two now when you're factoring by grouping that factoring by grouping
does not work of course unless you can get that group that expression that's in
the parentheses to show up again and if you can't get the X plus two in this
case to show up again then this polynomial is not factorable by grouping
and by the way polynomials are functions that have one independent variable and
all of the exponents are positive integers if you have a an exponent
that's negative or a fractional exponent then that's not a polynomial now we can
factor out between this common factors of 9 and 18 of course we have a common
factors of 2 and 9 you want to take out the biggest usually and we have these
terms of X + 2 which are positive our 9x + 18 or negative so yeah we're going to
factor out a negative 9 and negative 9x divided by negative 9 is positive 1x and
negative 18 divided by negative 9 again is positive - looking to make this sort
of basically less than or equal to 0 you noticed that we had that common factor
of x squared when we factored it out we only had it right at once we had a
common factor of negative 9 when we factored it out we only had to write at
once and here we have a common factor of x + 2 and we went from one two three
four terms separated by addition and subtraction to just two terms separated
by this one minus sign you have some pluses in here but the parentheses are
locking all that together like a vise grip so it is just two terms separated
by that minus sign so we're going to factor out that common factor of x + 2
and again we're dividing that out so x squared times X plus 2 divided by X plus
2 will just leave you with the x squared and negative 9 times X plus 2 divided by
X plus 2 is going to leave you with that factor of minus 9 now we have a
difference of squares that's a very common factoring pattern which I'm sure
by now you have down pat and a factoring of x squared minus 9 is going
come X plus three times X minus three just using some vocab out there this is
a pair this is a conjugate pair where the terms are the same but just the
middle sign has is different they're opposites then little signs opposite now
again remembering we're trying to think of a sketch that is going to help us
solve this polynomial inequality we're trying to quickly sketch y is equal to x
cubed plus 2x squared minus 9x minus 18 and in this sketch can even be in your
mind just a quicker way of helping you solve this polynomial inequality now how
do you get this all of this to equal 0 well any one of these terms could
potentially be 0 and this whole expression would be 0 so either X plus 2
is going to be 0 or it could be 0 X plus 3 or X minus 3 and of course the
solution to all of these one-step equations is X is negative 2 X is
negative 3 or X being equal to 3 any one of these X values is going to turn this
entire factor expression to be equal to 0 which makes them all x-intercepts of
this graph ok now this graph is not the solution well not this isn't the
graphing of the solution on the number line this is the mental or very quick
sketch you're doing on your paper to help you solve this problem so we have
that as X intercepts of negative 3 negative 2 and positive 3 great well now
what else do we know from graphing polynomials that's going to give us a
solution to this inequality without a whole lot of additional work getting my
offense a pitcher in pitcher the multiplicities the multiplicities of
these solutions these three solutions came from these three factors in each of
these three factors only showed up once which means their multiplicity is odd so
these x-intercepts of negative 3 negative 2 and 3
the graph is going through those x-intercepts and because our leading
coefficient is positive with an odd degree that combination is going to give
us an behavior of down to the left and up to the right so just with that little
bit of knowledge we can visualize this graph we have those x-intercepts of
negative 3 negative 2 and positive 3 we know the graph has an end behavior of
down to the left and up to the right and the graph is passing through these
x-intercepts now I guess it doesn't take too long if X is equal to 0 then my
y-intercept is negative 18 which isn't even on my chalk board here but the
graph is coming down its gonna come down somewhere down there at negative 18 and
come up through x equals 3 now in terms of graphs this expression the whole X
cubed plus 2x squared minus 9x minus 18 is giving our values of y if we were
graphing so where along this graph are my Y values less than or equal to 0
well that would include this portion of the graph and this portion up I don't
just going to show very marked very well against the yellow hopefully the red
sometimes red doesn't show up too much in the camera but those two intervals
are where the Y values are either equal to 0 or less than so with interval
notation our solution is between negative infinity coming in here from
the left and up until we get to negative 3 or from negative 2 to positive 3 then
of course that on a number line oh hold on a second I can actually include the
values of negative 3 negative 2 and positive 3 because we have an equal sign
as part of our inequality so that means I need to come back through here in my
interval notation and not use round parentheses which
means sort of just less-than or greater-than but include the inequality
which says no I'm actually including these numbers in my solution it's not
just going up to negative 3 or just between negative 2 and negative positive
3 we're not including those numbers now the directions say to graph this on a
number line so now we can just come through here and say there's 0 1 2 3
negative 1 negative 2 negative 3 negative 4 and our shading is coming in
from negative infinity up until negative 3 and then going between negative 2
solid dots because we're including those numbers again because of that equal sign
going on there and that's our graph on the number line that's in my first
solution we or example we got one more coming up right now already for our last
example very very similar to the previous one I'm just bringing this up
because we don't have an equal sign as part of our inequality and I just want
to remind you about that multiplicity again and what's the difference between
an even multiplicity and an odd so we have x squared minus 10x plus 25 is
greater than 0 we're going to solve this polynomial just like before let's get it
set equal to zero and we're gonna do that by factoring it we have a leading
coefficient which is just simply 1 and thus we're only looking for we have to
really consider it leading coefficient very much in our thought process of how
to factor this just look for factors of 25 that add to be negative 10 so we
don't want 5 and positive 5 we want to negative 5
x times X is x squared we have x times negative 5 and another negative 5 X
gives us negative 10x when you add them and of course negative 5 times negative
5 is positive 25 now I wrote each of those factors only once but they are the
same factor so I could write this as X minus 5 squared and again though this is
we're looking for an inequality words just great
than zero if we're thinking about graphing y is equal to x squared minus
10x plus 25 and using that graph to help us finish this solution of this
polynomial inequality then we need the x intercept so where is X minus 5 equal to
0 of course that's going to be x equals 5 we have here this x intercept the 0 is
coming from a factor which shows up twice now if you have an even
multiplicity that means you're just touching at this x-intercept and because
you have again a positive leading coefficient with a now degree which is
even instead of odd then the end behaviors remember when we were graphing
polynomial functions the end behavior was either up to the left and up to the
right or the leading coefficient was negative of course it was down and down
so knowing the end behavior from that positive leading coefficient or that
leading coefficient test and understanding that multiplicity here is
even so the graph just touches this is what the graph and it's just a sketch
I'm not going to make sure every single point of my sketch is really kind of
like in the proper place but I know I have a parabola where every single Y
value is you know above the x-axis except exactly at 5 where the y
intercept of the Y value is 0 because it's next intercept so our solution to
this polynomial inequality is that we have solutions from negative and
negative infinity to 5 now I'm gonna use a round parenthesis right because
there's no equal sign in our inequality it's just greater than or we can pick up
on the other side of 5 so like 4.9 mmm-ma and 5.000 0 is 1 and then we're
gonna go from 5 to infinity and beyond that's on the number line is going to
look like this and I forgot to mention this in the
other example we're gonna put an open dot on five because we're not 0 is not
part of the solution but we are going to show the shades show that the shading is
basically everywhere on the number line except for that open dot that we have at
five now the difference between either sketching very very quickly a sketch of
the polynomial or just even thinking about it in your head and the other
technique that we used for solving polynomial inequalities was that if we
were just doing this purely analytically this five would be sort of like a
critical number along the X along the number line and then you test not just a
single number in each that are in between each of those critical values on
the number line you take each of these numbers along the number line you know
one from each interval and you plug them into these factored expressions to see
if those answers are positive or negative and thus do they satisfy you
know the inequality what you're trying to solve but if you know how to graph
your polynomial functions and you know how that multiplicity works that
multiplicity here works with the X intercepts and your end behavior you can
get a picture of that polynomial your mind and use that or just get you
quickly for your teacher so you get you know some credit for showing your work
and allow you to speed up that process of solving these polynomials and
polynomial inequalities allowing you more time to work on the other problems
that are on your test and in your homework so I am mr. true hey go to your
homework
you