How To Type Squared Video
How To Type Squared Video Transcript
going on now to topic 5 of chapter 5we're continuing with polynomials and
polynomial functions now in this case
they've added the letter A as opposed to
not having it before because when a is
one it's like not having an a there but
now a is going to be not equal to one so
you're going to find maybe a two or a
three or four or five or any number
other than one at this position and
that's what we're going to work on now
there are no new vocabulary words but we
are going to move right into strategies
for solving when the a term is not one
so once again is it in standard form and
and can we take out any common factor
nope so again the strategy will be to
put two parentheses and again what we're
doing is we are undoing the foil so for
the first but now when you see a 10
there it could be a 10 and a 1 we know
there's going to be an X there so we can
put the X there if we want now this
could be a 10 and this could be a one
but that's not off in the case usually
it's going to be lets say a 5 and let's
say it to
now I have a difference now of one here
in the middle and over here I need to
what strategy are we using that is our
strategy guess and check so you're going
to do these in pencil because it might
not be the right one now if i put the
five here i'm sorry the three here
that's going to be a 15 that's not going
to work but i'm just trying this out
suppose I put the three here and the one
there ah see there's a six from my
inners and my utters give me a five the
difference is one ok now I need a
positive one here so the product of
three times to this will be the positive
and this will be the negative and how do
the 10 in the one because seventy
percent of the time it's not going to be
a 10 and a one so often you try the five
in the two and this is where it gets
more complicated and a lot more artisan
and figuring these out which is still
checked by foil all right as we look at
example 2 is there any common factor yes
there's a common a so we'll factor the a
out and this will then be 6a squared
minus 11 a minus 10 and now we put our
two sets of parentheses
now the a will go there and the a will
go there but as we do this it gets a
have a choice here for this one a one
and A six three and a two then we can
also have a two and A one and A six I'm
sorry a two and A three and a six in one
but we'll leave that here and remember
for ten you can have a one and a 10 and
A two and A five and that's where guests
and check comes in and then after why
you get a little bit of a feel now we
said before it was going to probably be
a 3 & a 2 so we'll try that and then we
look at our 10 well if I put a 5 here by
accident we got it yeah let's try a 5
there and at either because mentally I
saw that the outers we're going to give
me a 15 and the inners we're going to
give me a 4 and that gave me an 11 okay
that's what I'm looking for and my
middle sign is negative so the larger
number the 15 needs to be negative and
this one positive and if we foil it
that's a negative 15 a a positive for a
gives me my 11 a right there and then
when I foil it this way I get a negative
10 yep this is a good one so we got two
well done so far okay for number three
no common factor sort of in standard
form so well it isn't standard formats
soldier
parenthesis now we know our T's going to
go there and our T's going to go there
well let's work on the 40 there's a lot
of factors 240 let's see we might try an
eight and a five and we have a 15 is
going to be probably of 3 and a 5
somewhere or five and a three so I need
a 38 in the middle here so i put the
five there that's going to be a 44 don't
that's not going to work if I put the
three there let's just try it now this
is a 24 and a 25 nope that's not going
to work so we go over here to our eraser
then we erase it that's why you need a
racer guess and check
well let's try a 10 and a 4 again i'm
looking for a 38 there but if that's a
difference and so this one's kind of
tricky here ah look at this i put a 5
there and a 3 there my outers are 50 my
inners are 12 and fifty minus 12 is 38
so this is I think is the right
combination and let's see if we can
figure out what our signs are going to
be we need a negative so this will be
the negative and this will be the
positive and if I foil it yep again you
know sometimes you might have to switch
things around it's guess and checked
remember when we had easy things we were
just multiplying it was there and then
when a was one now have easily
challenging ones now as we look at
number four here is it in standard form
no so let's put it in standard form 20 x
squared plus 5x minus 15 now that it's
in standard form is there a common
factor yes we could factor a 5 out of
each term
and then still two sets of parentheses
now for number four now some of this you
can do mentally we could try a 2 here
and a 2 there and if we put a 3 that's
going to be six so we need something
with a difference of one so that's not
going to do it no matter where i put
three so a 2 into 2 i'm not going to use
i have to go to the 4 and the one oh I
see it down again we don't put this one
here but it's okay to put it there now
if I put two three there that's going to
be a 12 but if I put the three there
like so now when i do my otters i have a
four when I do my inners I have a three
ah so this is going to be a plus and
this will be a minus and that'll give me
that and that'll give me that and
that'll give me that okay good now I
hope I'm not going too fast but you know
we don't want to make the take too long
and again you can always pause the tape
work on it yourself and then see how
it's done so here common factor yes the
common factor of em em to the third so
that gives me then 35 m squared plus 9 m
minus two
okay there's no more common factor now
in here so we go m to the third two sets
of parentheses now pretty generally when
you have a number like 35 and your
middle term is nine and this is a two
it's not going to be 35 and one more
likely it will be five and seven I could
have put the seven there but doesn't
make any difference i see it already i'm
going to put the two there and the one
there why because this is 14 and the
outers are our five and the difference
is 9 so I need a positive middle term so
this is a positive 14 m and this is a
negative a 5m and the difference is a
positive 9 mm that's good and as I foil
it it checks out okay let's take a look
at this one here now any common factors
no so we're going to put two sets of
parentheses we know the X goes there and
the X goes there now for the four I have
a 4 here four and 14 and one could be
but again we say that's not usually what
happens it happened up there but i'm
going to try a two and A two because 16
and this is 15 now here I can put the
five here and I could put the three
there or vice versa it comes out the
same
ah very good look at this outers are ten
inners our six adds up to 16 so this is
it and they're all plus so actually this
was an easy one but again it's still
guess and check now as we look at number
seven standard form yes but notice our
first term is negative which we don't
like and I can take a two out of each
one so I'm going to take a negative 2
out of each of these so that will give
me a positive 9 see a negative 12 C and
a positive 50 that's going to be a
negative vibe going to change those
signs so this became a positive negative
negative i factored out it too good so
we still include the negative 2 in our
answer and this is Z squared okay so z
and z now we know this is going to be a
5 and a 1 somewhere but are we going to
use a 9 I think it's going to be a 3 and
a 3 we always try that first and then a
5 here
so this is so there it is yep this was
kind of an easy one two so there's a 15
and there's a 3 difference is 12 we need
a negative in the middle so the larger
one which is our outers need to be the
negative this will be the positive now
the same thing over here we want to line
this up we'll factor out a negative 1
this now becomes 25 x squared plus 20x
minus 12 and I'm pretty sure this is
going to be a 5x and a 5 X here now
here's where I have my 12 and there's
more it's a 1 in 2 12 but two of the six
of 3 and a 4 I need a 20 in the middle
so let's see how I see it six and A two
because this is 10 this is 30 I'm
looking for 20 that's the difference now
for some of you as you do it it's going
to take longer because you have to study
it and work on your factors but now we
need a positive 20 so my positive outers
which will give me a positive 30 and a
negative 10 gives me my inners okay here
in number nine I can factor out ay ay
and Abby from each term and that then
gives me a
a squared minus 2a minus 15 so Rav stays
there now two sets of parentheses and
again i have a 2 in the middle of 15 8
this is probably going to be a for a & a
2 a and my 15 is going to be maybe a 33
here yes and a 5 there I had mentally
thought already of what these would be
but there's a 10 and then there's a 12
and the difference is too okay so let's
see what our signs are going to be our
middle term is negative so the larger
number in this case the 12 needs to be
the negative and then our inners would
be a positive okay now in number 10 are
they in standard form and the answer is
no so we have to put it in standard form
no common factor
and this is going to be a 2x here since
that's prime now factors of 23 23 is
prime so you only need there's only one
set of factors now if I put 223 here
that doesn't give me a one if i put the
23 there that doesn't give me one so
this is not factorable this will be
prime now remember sometimes they you
know you got to give it a try and I
could have figured probably this one out
with that 23 since that number is prime
there are no factors that are going to
give you a one in the middle now the
purpose of this video is to give you a
modus operandi a method of operation the
strategy for factoring if I do every
example and it's going to make the take
too long so again here we see you could
take a 3 out of each term at a T squared
that would be my hint remember the
answers are in the back of the chapter
here ah this one's a little different
because this has a P squared and a Q
squared well it's done exactly the same
way and this one looks like it might be
yes it is a perfect square trinomial
that we had mentioned earlier so let me
do 13 for you
now we know there's going to be a see
here and a see here and there'll be a d
there and addie there now notice this
term is a perfect square and this term
is a perfect square and this happens to
be the product of the square root of
this which is five the square root of
this which is two and that gives you 10
and it's doubled so this matches the
recipe of a perfect square trinomial so
I'm going to put a 5 here a 5 their 14
was similar to what we've done so let's
go on to number 50 now we did something
similar to this this is a quadratic
equation it factors into this but if we
want to solve it remember we're going to
equal this 20 so these are the factors
equal to 0 so to solve this remember if
it looked like this we just take this
term transpose it equal it 2-0 and it's
then in this form equal to 0 we factored
it and now we solve it so this is going
to be 10x minus 1 equals 0 or X plus 3
equals 0 and again we're using the
principle of zero products here so here
we transpose the negative 1 it becomes a
positive one and then we divide both
sides by 10 here we get x is equal to 1
10 and here we know x equals negative 3
and this is very similar again we're
going to equal this 20 it's this one
this is the factorization and we say 3x
plus 2 equals 0 or 5x plus 4 equals 0
and then this will become a negative 2
and then you're going to divide by 3 so
X is a negative two-thirds here this
becomes a negative 4/5 X is a negative 4
fist and we'll try 17 in just a moment
ok to save a little time I'm going to
already factored these and then I
transpose the five make it a negative
5/3 here we're just transposing the
negative one making it positive
second-degree two solutions now here
this is to the third degree so I had to
factor out the X which I then got this i
factored that and then i divided each
well some cases i did hear this x equals
zero here I transpose the one to the
other side made it a negative 1/3 here
I'm just transposing and I didn't do
that one very well so let me clean that
up a little and this just becomes a for
this equals 0 transpose transpose the
for the negative 4 becomes a positive
okay here i had to transpose the 15 put
it in standard form i factored it and
again you had to play around it didn't
look like it was going to be a 16 and a
one or a four and A Ford so i tried the
6 in the two and then the three and the
five and then my outers gave me a 40 my
inners
six the difference is 34 my signs so
again this equals zero this equals zero
i'm going to transpose the three that
becomes a positive 3 divided by the
eight so that's one value then i'm going
to transpose the five that'll be a
negative five then i divide by the two
and there's my other value i'm making
this going a little quickly here because
again i think the skill that you need
here is factoring and then applying the
factor to the principle of the zero
product which is what we're doing again
we've done now quite a few examples but
here's another as you look at it it
doesn't quite look like a quadratic
equation but again it's not in standard
form so we have to distribute this we
get that we have to transpose the seven
to the other side becomes a negative 7
now this is in standard form of a
quadratic equation which will then
factor and as we factored it these are
our solutions now for number 21 finding
the zeros well it's just putting a 0
there factoring it and again solving
frags now keep in mind earlier we had
these and we had to make it in took me
quite into an equation or into a
function so keep in mind how this worked
this is the value over there and this is
the coefficient of our X and its
opposite sign so we're going to be using
that probably soon we will skip number
22 that's not in our curriculum but and
probably won't have many like this but
what's the strategy here of finding the
domain of these functions
well we want to find out what value of x
in this denominator would make it a zero
so we just in a sense equal our
denominators 20 in both cases and then
we will find our domain okay to wrap
this up then we have equated our
denominators 20 and what I did is I just
transposed everything to this side so
this would be positive and our net r 5x
will be negative equal to 0 factored out
the 5x got these two factors equated
each factor 20 and i have here that
dividing both sides by 5 here x is 0 and
here x equals a positive 17 now those
are what x equals and if x were these
numbers it would make our denominator 0
therefore our answer for domain is the
domain is all real numbers but X cannot
be equal to 0 and cannot be equal to 17
these are the two values for this
denominator that are excluded because
these values make our denominator is 0
now if x 4 0 this is 0 minus 0 is 0
that's why it if you put one seventh in
there it does the same thing a little
more difficult to do it all right now
for number 24 again this is to the third
power so there's three answers I
acted off The X Factor this out got
these three factors equated each of
these factors to zero so this is 0 this
becomes a positive 2 this becomes a
negative 3 so again the domain all real
numbers but X cannot be a 0 a 2 or a
negative 3 now in the answer sheet i see
they show you how to put it in interval
notation that's fine we're not going to
be using that but that'll wrap up this
section there's only one lesson left in
math 100 which I hope to finish this
weekend all right I'm all excited