Hey! Got a minute? Good because it's time for... a One Minute Wednesday
One cache... One minute. Cue the title... and start the clock
Hello Dumbo
We're with a supporter of Cache Canada and your name is...
Scout Mom 5
And she watches us every week!
Hey look who we bumped into, this is Tony and Ashley on GeoTrail
We're on the trail and we met them here at the Geobash - hey hey - Canada all right
and don't forget to check out their channel it'll be a link on the bottom -
and maybe at the end - Cache Canada get the GeoTrail
and that's only one minute of the Midwest Geobash. Look for these guys
channels Kneel More Cache Tales and Geocaching with Halemeister for longer
versions of the Midwest Geobash you guys having a video on that yes
absolutely so we'll put a link to their channels at the end and down below and
thanks for watching one minute Wednesday and there's another one next week on
Wednesday on
Cache Canada!
Yayyy, see ya
Did you remember to look for the secret code? This is one of the videos for the
watch and win contest that could have a secret code in it. Either four letters or
four numbers check it out again if it had a secret code write it down watch and win
Okay is this is for Friday?
No it's for Wednesday
Okay I gotta take my sweater off
It doesn't matter
No, I started with my sweater off
For more infomation >> Geocaching - One Minute Wednesday - A MEGA Minute - Duration: 2:26.-------------------------------------------
Me and the Better half play: please love my computer game #maybe - Duration: 49:54.
so this is where we start
the man said it would happen like this
like waking from a dream to dream
dazed
wondering what is real
the clocks tell a time i cant read
all lights seem to be on but its so dark
i can hardly see my hand infront of my face
its cold
colder than it should be
these blankets are thin and worn
the carpet to my side its ragged and plucked
like strings
frayed
its lonley
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Finale Massenmigration offenbar beschlossen (von Eva Herman) | 01.08.2018 | www.kla.tv/12800 - Duration: 5:28.
For more infomation >> Finale Massenmigration offenbar beschlossen (von Eva Herman) | 01.08.2018 | www.kla.tv/12800 - Duration: 5:28. -------------------------------------------
HOW TO DOGFIGHT #2 - HOW TO GET OUT SOMEONE OF YOUR 6 O´CLOCK (DEFENSIVE) English Subtitles - Duration: 11:35.
he what happen panda!!!??!!! Welcome again to revientor reborn in the dogfight tutorials
Today I am going to speak about evasion, how o try to avoid that enemy behind you
the first situation is the one situation that you never never ever you have to do....
right now we are at 500 knots, and we can see the target in front of me, is Crashius, how are you?, HI THERE!!!!
again in this situation we see that Crashius is doing a sustained turn... all the time , you must not do that
because is very easy for me in this position to aim and finally shoot the target
and there is my shooting... versus that "steady"target...
is very easy to hit that target, that is the only situation that you must to avoid, "be steady" or predictable
other situation that is not good do... you can do it but could be good or not so good...
is perform this kind of combat at 700 knots... thee is no chance to overturn when flying at 700, only do that to escape from the combat
at this speed you have very poor turn radio... you will not defeat the enemy with that....
remember this is a video and is prepared, forget about that 2G load factor that you see now in your screen
please crashius change the turn
imagine that we are in 700 knots, is this kind of speed when changing the turn.. really easy to do not loose the track of the target...
that means the turn radio is so big that I am all the time inside the circle
inside the circle I can use less fuel and power to keep the track and shoot
in this scene we´ll see something that you must do, we are going to try to keep same speed,
and you must notice that Crashius in this case is maneuvering, and I follow him, he is performing that turn to force to me to maneuver and according my maneuver
and that can make that I do some mistake, each mistake is increasing his possibilities to survive and negate my shoot
as you can see now I am close, of course I can fix that using less power in the engine
the only way to escape in this "picture" is force to the enemy to overchute him...
now I am going to overchute...
because I was no paying attention to my speed finally Crashius force me to overchute...
and now is a neutral fight
because when I was in offensive I forget to check the target speed and my speed
that is the idea
Basically what you have to do is always looking the contact behind you, now I am going to change roles with Crashius
this is what you have to NOT DO NEVER
Wait a sec I am going to turn off the RwR audio
in this case, I am turning steady... very predictable ... you must not do this...
in this moment I know the target have not aiming on my
because I do not see the belly of the plane
but I can have shoot very soon. look the zoom,
as you can see you see the upper part of the plane, is not just in front like this
this is How I am watching the plane, this indication say hes is not able to fire guns Vs us
in the moment you see some belly that means he will soon shoot at you
now from the cockpit.... please Crashius...
there he can shoot at me
in that moment I have to change my posit ASAP
all the time try to change your altitude a few feet
then is more difficult to be shooted down
change the side of the turn when the target is getting shoot opportunity
finally all this movement are accumulative error in one plane or other.. to finally one of them make the big mistake an loose the game
you not have to way to see the belly of the airplane... you can change your heading any time, but always be unpredictable...
you can combine this movements, using the air brakes or slow down the airplane, to make the enemy to be closer and closer to finally break my energy to make the enemy overchute me
remember the airbrake is visible also the afterburner flame...
with the G load you can brake the plane
finally Crashius overchute me
this is a neutral fight and I can escape or fight
because he have move energy than me he got the vertical and again is at my six
that was the view from inside and outside
that is all you must know to take someone out of your six, every combat is a accumulative decision and error.. more errors you will loose
in this maneuvers at 500knots and 4gs, if you in that moment use IDLE , 9 Gs, changing the turn you are giving away your energy to force to the enemy to do something
the enemy is not all the time watching your speed, in that moment if he do not see that... you are getting what you want
remember to keep some energy you still in the bad position you need to have something to use and maneuver
normally if you overchute someone you have to go to the vertical with your excess of energy and then come back to the 6 of the enemy
someone that is not experienced in dogfight will try to brake more than you and finally he will overchute you with no energy and you will kill it
now we are going to perform the same as the previews one, but more realistic
everything afterburner, idle, airbrakes.... dirty stuff....
only no guns xD
try to be real.. a real pilot can not manage all the time 0 to 9 Gs movements...
and there is the neutral position
I lost visual two times, that is important...
as you can see Crashius was overchute but manage to get 2 more shoot opportunities
that is all,
I hope you liked today´s video, as you saw there is no much tactics in there... may be a few more tips but all the time is overchute thing
of course you can do the turn in the vertical or other axis...
in this case train that, be at the six of someone and the other scenario, have someone at your six...
the common mistake is use afterburner and forget you speed , remember the rudders and the airbrake
subscribe, thumbs up, commend below, tips or things that you need, thanks Crashius to be here, bye bey and be happy!!!
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Cristiano Ronaldo (Juventus) vs Luka Modrić vs Kylian Mbappé Age Transformation Who Is Better?? - Duration: 4:23.
Cristiano Ronaldo (Juventus) vs Luka Modrić vs Kylian Mbappé Age Transformation Who Is Better??
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Nightcore - Stereo Hearts (Lyrics) - Duration: 3:43.
My heart's a stereo it beats for you. So listen close hear my thoughts in every note
And make me your radio turn me up when you feel low
This melody was meant for you sing along to my stereo
Honey got a sexy Asti men. Give a hotness a new meaning
Perfection. Mommy you claim then?
Inception you got a brother dream and dream in. Yeah, baby. I'm fiending
Trying to holla at you. I'm screamin. Let me love you down this evening
Let me love you love, you know
you are my teen night and only got a note to make you understand think it's awfully kinky and grab me by the
Hands and keep me stuck inside your head. I'm like your favorite tune and know my heart is like it's never your place view
My heart's a stereo it beats for you. So listen close hear my thoughts in every note Oh
Make me your radio. Turn me up when you feel low
This melody was meant for you sing along to my stereo
Today I don't feel like doing anything I
Just wanna lay in my bed
I
Don't feel like picking up my phone to leave a message at the tone today. I swear. I'm not doing anything
If I could only find a note to make you understand singing softly you can't grab me by the hands
You keep me stuck inside your head like your favorite tune, you know, my heart is like a stereo a place for you
My heart is there beats for you so listen
in my thoughts in every note well
Make me your radio
Turn me up. When you feel low. This melody was meant for you sing along to my stereo
My heart's a stereo
It beats for you. So listen
in my thoughts anew Oh
Make me your radio. Turn me up when you feel melody was mentally
Long Zuma stereo, my heart's a stereo. It beats for you
So listen, close hear my thoughts in every note. What Oh,
Make me your radio
Turn me up. When you feel low. This melody was meant for you sing along to my stereo
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VIP Protocol Imran Khan Rok Sako to Rok Lo Tabdeli Ai hey - Duration: 2:25.
VIP Protocol Imran Khan Rok Sako to Rok Lo Tabdeli Ai hey
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[ MATCH 105 ] du FIVE SONGS - Duration: 5:46.
For more infomation >> [ MATCH 105 ] du FIVE SONGS - Duration: 5:46. -------------------------------------------
Darısı Başımıza Final - Duration: 1:07.
For more infomation >> Darısı Başımıza Final - Duration: 1:07. -------------------------------------------
أغنية شارة محققو الحيوانات مع الكلمات - Duration: 1:45.
For more infomation >> أغنية شارة محققو الحيوانات مع الكلمات - Duration: 1:45. -------------------------------------------
Light eggplants parmigiana - Duration: 3:18.
Hello, hello Welcome back to Smart Cooking
today we prepare the eggplant parmigiana
and Silvia lists the ingredients
So, first we need the aubergines
that we have prepared previously
Click here for the video
for those who do not see the video are baked aubergines made with oil and salt
and that's it, they're light. I do not mean diet because t here is oil
but it's not so much so it's okay anyway
then we need parmesan
we have already scratched it
Mozzarella
and the tomato that sofia has done before
a simple sauce that I seasoned with a little oil, salt and basil
that's all, then we assemble and show you how
In a 180 ° oven
well, our parmigiana is ready
and now we taste
cut? Cut it
with much happiness
I was hungry
much
oh mom, good! It went out
good, very delicate
goodbye everyone, we have to do
If you liked this video Subscribe to the channel and put Like
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Segovia, Spain: Architectural Beauty - Rick Steves Travel Bite - Duration: 2:25.
Segovia, an historic town of 55,000
with a rich history and a famous Roman aqueduct
is well worth a short visit.
Nearly 2,000 years ago,
Segovia was a Roman military base in need of water.
So Emperor Trajan's engineers rerouted a stream into town
by building this nine-mile-long aqueduct.
This massive structure,
ingeniously constructed with no mortar,
provided water to Segovia until modern times.
While locals claim it could still work today,
its primary function seems to be
providing a backdrop for family fun.
Mommy,
mommy!
Segovia's main pedestrian drag
is clogged with people enjoying
a festive spring day out.
While Romanesque churches and statues
honoring 16th-century local heroes are ignored,
kids get plenty of attention.
[Speaking Spanish]
The Plaza Mayor marks Segovia's old town center.
It's early in the afternoon on a Saturday,
and, as on plazas all over Spain,
people are out with family and friends.
It's prime time to enjoy an aperitivo
before heading home for the main meal of the day,
a late lunch.
Segovia's Plaza Mayor wasn't always so idyllic.
It was long the scene of bullfights.
Imagine spectators jamming these balconies, screaming,
"Toro, toro, olé!"
In the 19th century, bullfights in this square were stopped.
When the residents complained,
town fathers gave them a more gentile form of entertainment:
this bandstand.
The town's cathedral stands high above its main square.
Embellished to the hilt
with pinnacles and flying buttresses,
it's a good example of that final overripe
stage of Gothic called "flamboyant."
[ Spanish guitar music ]
Segovia's Alcazar, or fortress,
still seems to defend the far end of town.
Once the king's summer retreat and palace,
it burned down and was rebuilt.
Like so many castle rebuilds
in the romantic 19th century,
it's a fanciful exaggeration,
barely resembling the original.
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VENOM TRAILER 2 | ORIGEM DO SIMBIONTE RIOT - CENAS ANALISADAS - Jujuba ATÔMICA - Duration: 12:31.
For more infomation >> VENOM TRAILER 2 | ORIGEM DO SIMBIONTE RIOT - CENAS ANALISADAS - Jujuba ATÔMICA - Duration: 12:31. -------------------------------------------
The Most Dreadful TV Shows Of 2018 - Duration: 5:50.
Even a few years after the conclusions of all-time classic shows such as Mad Men, Breaking
Bad, and Parks and Recreation, we're still in "The Second Golden Age of TV," also known
as "Peak TV," also known as "There is Far Too Much Good Stuff to Watch on Broadcast
Television, Cable, and At Least Four Streaming Services."
There's just so much to watch: Game of Thrones, Westworld, The Walking Dead, Atlanta, Silicon
Valley.
Still, despite all this can't-miss TV, there are plenty of other series you can go ahead
and skip.
The following are our picks for the very worst shows of 2018… so far.
Alex, Inc.
ABC promoted Alex, Inc. as the triumphant return of Zach Braff, away from TV for almost
a decade after Scrubs finished its nine-season run.
But it didn't quite work out that way.
This is partly a typical family comedy about a goofy dad, his long-suffering wife, and
their kids.
But it's also about a guy who has a midlife crisis, starts his own company, and documents
that journey through podcasting.
Based on the true story of StartUp host Alex Blumberg, Alex, Inc. reveals to audiences
the not-very-visually-entertaining world of people talking into microphones.
While podcasts are mainstream at this point, the show treats the concept like it's cutting-edge
technology, and people treat Alex like he's a genius for making one.
Simply unbelievable.
"I hate to disagree with you.
I hate it enough to make two fists."
Child Support
Ricky Gervais is a TV legend for co-creating and starring in the original British Office,
as well as the show business satire Extras and the nursing-home dramedy Derek.
But his latest project isn't a single-camera comedy or nuanced character study.
Instead, Child Support is one part Kids Say the Darndest Things, one part overly convoluted
game show.
In a TV studio, host Fred Savage asks adult contestants trivia questions.
If they answer correctly, their responses are compared to the answers given by a panel
of adorable children.
Gervais asks the kids the same questions in a creepy-looking focus group.
It's more unusual than amusing, and as The Laughing Place has noted, the questions are
"painfully easy" and the show "tries too hard to be funny."
"Benjamin Franklin said, 'Early to bed and early to rise makes a man healthy, wealthy,
and' what?"
Deception
The early 2000s are over, but that era's biggest contribution to television endures: the crime
procedural with a sassy twist.
The likes of Bones, Castle, and The Mentalist are all gone, but they continue in spirit
with ABC's Deception.
It's about a celebrity magician with the very celebrity-magician name of "Cameron Black."
After his career is destroyed by a scandal, he teams up with the FBI to solve crimes.
That's right, he uses his gift for "illusions" to somehow get into the minds of criminals,
which is a pretty weak premise.
But winning over viewers is one magic trick Cameron has not managed to pull off.
Add in meaningless supporting characters, and what's been described by Variety as "a
fatal lack of surprises," and there's not much left to watch.
Life Sentence
The breezy CW dramedy Life Sentence offers up basically the same setup as the 2001 Disney
movie Max Keeble's Big Move.
But in this case, it's about a young woman named Stella who thought she was dying of
cancer.
For years, she lived everyday like it was her last, only to have her cancer cured, leaving
her to deal with the ramifications of her impulsive decisions.
There's a decent premise here, but the execution just feels too cruel to its main character,
while leaving other moments overly sentimental.
Ultimately, it's just an uneven mix.
Living Biblically
Based on humorist A.J.
Jacobs' book The Year of Living Biblically, this CBS sitcom is about a critic who decides
to live out every last detail of his life according to the Holy Bible.
While that's a refillable premise, it's difficult to make religion funny and palatable to a
broad audience.
Secular audiences are alienated, while the show also skirts offending the faithful, never
really finding an audience.
Ultimately, Living Biblically commits the cardinal comedy sin of just not being very
funny.
The Resident
Matt Czuchry stars as Dr. Conrad Hawkins, a medical rogue who plays by his own rules.
Dr. Hawkins makes rash decisions and ignores directives from his superiors because he has
gut feelings about what crazy medical procedures to use.
Except he doesn't come across as heroic; just arrogant and smarmy.
Instead, he's just another antihero who thinks he can get away with anything like we've seen
so many times before.
And that is The Resident's biggest flaw: it doesn't offer anything new to the TV landscape.
Sweetbitter
This Starz series follows naive Tess, who leaves her hometown behind to move to the
Big Apple.
New York is unexpectedly tough, until Tess is taken under the wing of colleagues at the
fancy restaurant where she finds work almost immediately.
She reminds them of themselves, only younger and more optimistic.
Except cliches are a big problem on Sweetbitter, a very slow-moving show where nothing much
happens beyond a woman working in a restaurant and making friends.
Robert Lloyd of The Los Angeles Times called it:
"A meal that looks great on the plate but makes little impression on the tongue."
Splitting Up Together
This ABC family sitcom is at least an innovative, if not vaguely depressing, twist on an old
format: Mom and Dad are in the process of divorcing, but rather than have one of them
move out, both parents stay in the house, keeping separate living quarters so as to
still provide their kids with an uninterrupted and stable home life.
While that's a modern and progressive solution, it's a hard sell for a silly little sitcom.
And according to the Orlando Sentinel,
"It feels like a long, forced push to get [the main couple] back together."
Here and Now
Set in the progressive city of Portland, Oregon, Here and Now is about a progressive family
made up of a philosophy professor, his therapist wife, and their four mostly grown children,
adopted from around the world.
Their one biological child is in high school, where she encounters many didactic conversations
about social issues, offset by weird dreams and prophetic hallucinations.
Basically, imagine if Portlandia wasn't a comedy, but was instead just as serious and
ominous as The Leftovers.
According to Joel Keller of Decider,
"Here and Now is so overwrought that it makes The Handmaid's Tale look like a light-hearted
romp by comparison."
Dark.
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"Ghajinikanth" Premiere Show | Special Show | Arya | Sayesha | Santhosh P.Jayakumar | Kalakkalcinema - Duration: 2:31.
For more infomation >> "Ghajinikanth" Premiere Show | Special Show | Arya | Sayesha | Santhosh P.Jayakumar | Kalakkalcinema - Duration: 2:31. -------------------------------------------
How to Dye Faded Jeans - Duration: 4:46.
Hey guys, it's Clara from Online Fabric Store.
If you have a faded pair of jeans that you want to revive, why not dye them?
I'm using denim blue Rit dye on these, and black Dylon dye on these. So let's get started.
The materials you'll need are: jeans, Rit dye or Dylon dye,
For Rit dye, dishwashing soap and Rit color stay fixative are also recommended,
salt, measuring cups, rubber gloves ,and a container for dyeing.
Machine wash the jeans in warm water before dyeing.
You can use either type of dye or try a different color for a unique look.
Cover your work surface with plastic and wear rubber gloves when handling the dye.
One packet of Dylon permanent dye dyes up to half a pound of fabric.
These jeans are a little under a pound when dry so I'm going to use two packets
Dissolve the powder in four cups of warm water. Be careful not to inhale the powder
Stir until the dye is completely dissolved
Fill a container with as much water as needed for the jeans to be able to move freely
The water should be warm. Add the dissolved dye and stir thoroughly
Mix in a third of a cup of salt
Submerge the jeans and stir constantly for 15 minutes
The dye looks blue, but don't worry. It'll turn out black in the end
Keep in mind that many jeans contain a percentage of polyester or other synthetic materials, which don't dye
You may not be able to get a super dark black if this is the case
Let it sit for 45 minutes starting occasionally. Rinse the jeans in cold water
Machine wash in warm water and air dry away from direct heat in sunlight
Wash separately the first couple of times to remove any excess dye
Fill a large plastic container or stainless steel pot or sink with three gallons of hot water
If your tap water doesn't get scalding hot heat up the water on the stove until it's near boiling
Add 1 cup of salt to enhance the color and about a teaspoon of dishwashing soap to help the fabric dye evenly
If you want the darkest results possible keep the water on the stove the entire time to maintain the heat
Use half a cup of dye or if you want a darker color use a full bottle. For light colors use less dye
Shake the bottle and pour it into the water
Stir well before adding the damp jeans
Stir continuously, especially for the first 10 minutes
Make sure the dye is getting to all parts of the jeans
For dark colors leave the jeans in the dye bath for at least 30 minutes while stirring often
The denim fabric used for jeans is typically woven with both indigo and white threads.
This gives denim its characteristic look
When you dye denim, it will create a flatter color
Use Rit color stay dye fixative to enhance the color and reduce bleeding.
Fill another container with about 3 gallons of hot water
Use half a cup of fixative for every pound of dry fabric
Put the jeans in the fixative solution and stir for 20 minutes
Rinse in cool water
Machine wash on warm and dry
Wash separately at first then with like colors
Now that the jeans are dry you can see how the colors turned out
The black isn't super black but it's much darker than before
The other pair turned out to be a nice dark blue
The denim still has its characteristic look because the white threads didn't dye much meaning it's probably polyester
The results will differ for other genes depending on the percentage of cotton and the starting color
But that's part of the fun of dyeing. You never know how it's going to turn out
Thanks for watching this OFS project
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Search warrants reveal more about deadly crash - Duration: 1:48.
For more infomation >> Search warrants reveal more about deadly crash - Duration: 1:48. -------------------------------------------
How to Answer the Age Question when Dating Younger Women - Duration: 4:43.
Hi! my name is Eric Michael, founder of the Power Life System and you're gonna
be learning specific things to say when you're dating younger women and the
question of your age comes up this is the second video in this series and the
first one focused on foundational concepts because when you have your
foundation you'll believe systems your concepts right then you can say many
different things I'm gonna give you some examples of specific things one can say
however you're gonna be able to customize these or have things come into
your mind when you have your belief systems in your foundation right so
there's a link below the video I highly recommend watching that first and then
continuing with this video so the first example is is that somewhere along in
the conversation she asks how old are you and you respond saying how old do
you think or what would you guess my age is and then she's gonna think maybe
she'll say I don't know if I can guess I don't know I can't guess you said no go
ahead just guess any numbers fine she's like guessing younger than your
age you can say I love you you're great and give her a hug if you want and then
continue the conversation many times she's not gonna ask again or if she does
ask again you can simply say what your age is and she's probably gonna look at
you say oh my goodness you look a lot younger go say thank you and continue
the conversation normally I mean there's a million different things we can talk
about as a second example somewhere along the conversation she
asks how old are you one way you can actually answer that is by exaggerating
you tell her you're 85 years old but I use a lot of vitamins or skin cream and
you got to make a joke out of it you're laughing at yourself and then sometimes
she's not going to she'll laugh at that too to continue the conversation and
you're kind of downplaying the question in a sense too however she may ask well
so how old are you really at that point should answer you say whatever your age
is then she may say oh well you look a lot younger you say thank you continue
the conversation third example she asks your age if she guesses higher than
again watch the video that I have below that's an introduction to my course
greater health gains some valuable information so that maximize your look
and your appearance so you look the best regardless of
whatever your birth date is then she's gonna guess says she guess his younger
and you said I love you that's great how you're the best you were high five hug
her if you want and then if she asks again you tell her your age and you say
whatever your age is and likely it's gonna be older now we're gonna kind of
make fun of ourselves this is really good if she had guessed younger and
you're actually are in pretty good shape or if you have the personality you
congruent with it being playful and I mean I'm making fun of yourself because
you are so confident where you can tell her and you can grab her armor you're
gonna need to walk me across the street and I have my walking stick and likely
she's gonna roll her eyes and no it's kind of ridiculous what you're saying
because obviously you're in you're in decent shape and then you continue the
conversation I've given you a couple of examples you can customize them I want
you to look more at the concepts and also watch the foundational video the
best way to solve this question in general of how old are you is to have
all areas of your life Rocking financial spiritual enjoyment social everything
because ultimately when you have that you're so attractive that this becomes
it's called total moot point even if it does come up it's not important at all
and you're naturally just going to continue the conversation it's no
different than oh so what's your favorite restaurant to help you do that
I've created a free life abundance quiz it's free so click below the video you
can do it right now and I'm going to give you your free results as well as
video guidance to help move you in the direction of where you most want to go
so you can have every area of your life Rocking
my goal has been to help you in what for many men is a precarious situation and
really put you in the right place frame of mind so you can ultimately have the
greatest connection with the individual you most desire to have it with and
create wonderful experiences for her and for you and together so I thank you
ahead of time for giving the video a thumbs up subscribe to the channel so I
can share lots more with you and help it possibly impact your life and I think
you ahead of time for adding a comment below the video of something that you
liked about the about this what I've been sharing as
well is something maybe a question you have or what you'd like me to go even
deeper with so that I could help you even more
I send you lots of great energy look forward to hearing about your success
story with this individual you most desire and have a fantastic day
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Horner's Method - Duration: 9:57.
hello Internet Oscar Veliz again this time with a video on Horner's method
first we'll go over a brief history of it then talk about how to evaluate
polynomials and lastly how to find roots Horner's method is credited to William
George Horner who published a new method of solving numerical equations of all
orders by contains approximations in 1819 although the method had previously
been found by Paolo Ruffini in 1804 they actually had been discovered even before
then and I'm gonna apologize right now for my pronunciations of names in the
11th century al-Nawasi created his own method similar Horner's that solved for
cubic roots and then in the 14th century al-Kashi extended that to solve for roots
of higher degree al-Tusi and al-Samaw'al also figured out their own approaches that
are similar to Horner's method for solving nth-roots around this same time
in China Qin Jiushao published a treatise in 1247 that had a way similar
to Horner's that could solve roots up to 10th order earlier in the 12th century
another al-Tusi wrote a book that also included something similar Horner's
method going back to china in the 11th century Jia Xian wrote a book that's
been lost to history but is also supposed to have included Horner's
method there's also evidence that Yang Hui Li Ye and Zhu Shijie again so
sorry for my pronunciation also solved polynomials in a way similar to Horner
and probably the earliest example of Horner's method comes from chapter 4 of
the nine chapters on the mathematical art which is an ancient Chinese text and
there are likely other examples of Horner's method being discover earlier
but to make a long story short I'll leave you with this quote that says both
Ruffini and Horner were blissfully unaware that they were rediscovering a
method which could be traced back 17th centuries before we start talking about
the method let's first define what we knew at a function and the polynomial so
a function is any function that you can think of mathematically something like
sine of e to the X plus the natural log of 1 over X square plus 1 while a polynomial
is specifically an addition and multiplication of powers of coefficients
so something like d x cubed + c x squared plus b x plus a you've seen these a lot
when we evaluate polymer this can be slow you'll notice that the last
operation requires three editions and six multiplications so every X to the N
term ads n more multiplications the idea behind Horner's method is that
we'll rearrange our polynomial so if our polynomial looks like this we can factor
out an X from the last two terms and factor out an X again this leaves us
with three additions and three multiplications so it's a lot faster it
if we write it in terms of a sub n you could represent it normally like this or
rewrite it in terms like this recursively it will look something like
this let's look at an example this rewrite and evaluation process if we
take this polynomial will first rewrite it then if you want to evaluate the
polynomial let's say three we'll simply plug in three where we had X's then we
start evaluating the polynomial for this example P of 3 is equal to 960
let's evaluate that same polynomial at another value this case - we'll take
our polynomial and plug in 2 now I'm not going to show every operation here but I
want you to keep track of these numbers first 1 -6 -84 214 1155 and finally
0 so P of 2 is equal to 0 this means that 2 is a root you may see Horner's
method represented as synthetic division where we take our polynomial and keep
track of the coefficients and evaluate it at a certain value
here's how we'll do that first we take our value and put it on the left side
then write all of our coefficients next to it
first we'll bring down the first coefficient then multiply that
coefficient by the value we're searching for this case two and store it next to it
then we do addition and repeat the process so now I'll multiply this -6
times two store it to the right and we get -12 then we add it again
and so on and so forth notice the same numbers 1 -6 -84 214 155 and
0 so we know that P of 2 is equal to 0 but we also found the quotient Q with
X to the fourth minus 6 X to the third minus 84 x squared plus 214x plus 1155
this is a basis of Ruffini's Rule we evaluated a polynomial at the value 2
what we did was essentially divide that polynomial by X minus 2 and we found
these numbers they came from this if we took our polynomial and rewrite it in
terms of its roots we divide out that X minus 2 and we're left with these values
which we can expand to be the quotient that we found Ruffini's Rule says that
every polynomial is a quotient times a binomial plus a remainder so for example
polynomial that we've been using our first example was 3 so we write x minus
3 and a remainder of 960 and this was our quotient notice that P of 3 is equal
to 960 with P of 2 we found this quotient X minus 2 and remainder of 0
notice that P of 2 is equal to 0 so P of at whatever value a is equal to our
remainder and this is the basis of ruffini's rule that P of X is equal to Q
of X times X minus a plus P of a Ruffini's Rule also lets us compute the
derivative of a polynomial so if you start with Ruffini's Rule and take
the derivative of both sides we expand it using the product rule then we can
simplify this since P of a is a constant so the derivative that is 0 which we can
then simplify further notice the still requires another derivative the
derivative Q if you plug in a now we have a minus a which simply gives us the
derivative P at a is equal to Q at a here's an example the derivative process
in action let's use our same polynomial and compute its derivative using the
power rule which is this in which we can simplify to that polynomial there if you
evaluate at say the value of 2 we're given the value of 1215 if you use
Ruffin's rule instead we compute the quotient then plug in 2 and this gives
us the value of 1215 using what we know now let's talk about finding the roots
of polynomials in three different ways the algebraic way the rational root
theorem and using Newton's method the algebraic way is quite simple and
you've probably done it if for our polynomial second order we can use the
quadratic equation if it's third order we can use the cubic equation which has
a little bit longer discriminant if it's fourth order the discriminant becomes a
lot longer and if it's fifth order there actually is it an algebraic way to solve
it this is actually in the proved by the Abel-Ruffini Theorem we could instead
use the rational root theorem which says given a polynomial we test that
polynomial at every combination of values of the integer factors of a sub
zero divided by the integer factors of a sub n both positive and negative
for example polynomial all this means all the combinations of the integer
factors of 2310 divided by the energy factors of 1 which gives us these 64
test points which actually can find the roots but as the name implies this won't
work for irrational roots and the larger your numbers the more test points you
have [assuming not prime] let's instead use Newton's method but apply it to polynomials if you
haven't seen my video on Newton's method I recommend you watch it now since we're
using Newton's method we'll need the derivative which you can find using
either the power rule or ruffini's rule and whenever we evaluate a polynomial
you'll want to use Horner's method as for starting points if you have a nice
mixture of positive and negative coefficients this means your roots are
they're all negative or positive and negative so you can start at zero as a
good test point otherwise you might try using the an through of the largest
coefficient or use one of the points from the rational root theorem
whatever the case once you do find an answer from Newton's method try rounding
it with the polynomial to see if that gives you a better result then once
you've found a good enough root deflate the polynomial using ruffini's rule and
restart the process let's take our example polynomial find the derivative
in this case we'll use power rule and plug it into Newton's Horner method
starting at zero the first iteration gives us 3.17 the
second 0.86 the third two point something until we get to
1.999..98 in this case we'll actually want to round the
root to be two because that'll give us a better answer
then we'll deflate a original polynomial using the root that we found in this case
we'll make the quotient that we found using two be our new polynomial restart
the Newton-Horner method find a root of -7 deflate again find the root
deflate find the roots deflate find the root this gives us all of our roots of
-7 -3 2 5 and 11 some things to keep in mind the coefficients
that you're using don't have to be integers they were just so for our test cases and
also Newton's method can still diverge but you get quadratic convergence every
time you find a root you also can compute P of a and Q of a at the same time for
efficiency and this method will only find real roots if you need complex
roots you can use Muller's method or Bairstow's method and the code that I
used will be hosted on github sincerely thank you for watching if you have any
questions or comments about the video please be sure to leave them below or if
you have suggestions for future videos I always enjoy hearing from you
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