You understand that students don't truly understand until they care. Thank you so much for always explaining the significance of what we are learning! You remind me of my high school Spanish teacher who ignited my passion for both languages and teaching. Now I'm back in school in my 30s relearning algebra so I can progress to calculus, and you have given me so much confidence.
Happy to see your still doing this. I solely clicked on this video because I was feeling a bit nostalgic. You're an awesome human being and your videos helped me greatly while taking calc 3 this past summer. Keep changing the world (and math education) for the better. :)
@@somasahu1234 I think it's because when you have -x as an input you're telling the function "Hey, what do I get out when I'm at the negative part of the x-axis?" And thus, your resulting polynomial is the output of that evaluation... It's kinda the same idea of Even vs. Odd functions, when you evaluate -x you do it to see what you get out when your input is any negative number in general At least that's how I interpret it
Astonishingly well explained. What could seem like a mouthful in the first 2 mins, becomes crystal clear minutes later and is consolidated through the examples.
Fantastic justification for why we subtract 2 from both +ve and -ve x-intercepts. Trying to teach in this way with a lot of patience is just amazing. Prof. you are simply awesome :))
Professor Leonard ,thank for smooth explanation of Descartes Rule of Signs. This is a classical way to determine the number of positive/negative x- intercept of any polynomial.
Thanks for this. It explains this concept in an incredibly easy to understand but concise way. It's not that hard of an idea, but thorough explanations help me retain, and I couldn't find one until now.
I'm looking into getting my sibling into Professor Leonard for math but I am Canadian so I'm not sure what the order of subjects is for his math lessons. If someone could tell me, it would be very appreciated. In Canada, in gr 12 we start with Advance Funtions into Calculus and Vectors. Then in university we do Calc 1. Of all the math playlist the Professor has, waht order should a young student watch them in? Thanks.
So quick question. Whenever we have a cubic function, and we do our real zero test, and then we do Descartes rule of signs, what comes after. So if we have 3 positive real zeros or 1 real zero, and 2 negative real zeros or 0 negative real zeros,how do we know which ones to test first with synthetic division
Hello Sir; thanks so much for the lesson. Please am a level hundred BSC. mathematics student. I will love to become a good mathematician like yourself. Need your guidance please. From Ghana
Okay, look: Imagine you have a polynomial given to you in a factored form, so say: f(x)= -2(x-7) * (x+1)^2 * (x+4)^3 * (x^2+9)^2 You can either distribute to find the highest power or degree of your function or do the fake distribution trick he showed us, either way it is going to be x^10. So we could have 10 intercepts. But the factored form of course shows us exactly how many we have: 3. For every factor but for the last one there is a solution. But you notice that the last one does not yield two solutions because: (x^2+9)^2=0. /square root x^2+9=0. /-9 x^2=--9 --> We cannot now take square root. So what we have lost in this factor is the possibility of two additional solutions. If the expression were x^2-9 we would have had -3 and 3 as additional x intercepts, right? Every single x raised to an odd power will have an x intercept: x^3 or x^9 or whatever, they have to go through once. So you know you'll not lose any with them. But with even powers there's the possibility of losing either none (they yield real solutions) or strictly two. When you get the standard form of that function, the x^10 is explicitly given so you know you could have 10. But since you have not neatly factored it and you don't know how your powers will be distributed across the fractions, you do not know which factors will yield solutions and which ones won't: So we have to assume that some of the factors could be such as to be irreducible, and since we don't yet know how many, we account for that by simply always assuming that we'll be losing two solutions. That's why we then have a list of possible x-int, depending on how many factors will turn out to deny solutions by being irreducible. Hope this helps!
complex solutions means it never touches x axis ... that means we need to take that into account... so subtract all even possibilities out of the counted no of sign change..... in case of odd I think at least one real solution will be there as Odd function is a journey from +∞ to -∞ or vice versa in North -South Direction ( Y Axis) .. it must pass X axis at least once during its journey via Y axis... where as an even function can totally bypass X intercept by flying high.. something like " complex functions are out of the Radar of X intercept " just plot f(x) =x^2 +36 and see it for yourself...
Because you always get complex roots in pairs (one complex number and other it's conjugate), so if a polynomial is of degree n, then either you get n real roots or you get some complex roots, since complex roots always occur in pairs, there can be either 2 or 4 or 6 of them and that means you get either n-2 real roots, (n-2)-2 or n-4 real roots and so on, that's why we always subtract 2
You understand that students don't truly understand until they care. Thank you so much for always explaining the significance of what we are learning! You remind me of my high school Spanish teacher who ignited my passion for both languages and teaching. Now I'm back in school in my 30s relearning algebra so I can progress to calculus, and you have given me so much confidence.
2 videos in less than 24hrs, this is amazing.
its not, dumb he already made that videos, wtf
Happy to see your still doing this. I solely clicked on this video because I was feeling a bit nostalgic. You're an awesome human being and your videos helped me greatly while taking calc 3 this past summer. Keep changing the world (and math education) for the better. :)
Wow, thank you!
You’re truly welcome!
@@ProfessorLeonard why f(x) for +ve x intercepts and f(-x) for -ve x intercepts ‽
@@somasahu1234 I think it's because when you have -x as an input you're telling the function "Hey, what do I get out when I'm at the negative part of the x-axis?" And thus, your resulting polynomial is the output of that evaluation...
It's kinda the same idea of Even vs. Odd functions, when you evaluate -x you do it to see what you get out when your input is any negative number in general
At least that's how I interpret it
I'm so grateful for the back to back uploads im currently taking Pre-Calculus and your videos help so much!!!
Astonishingly well explained. What could seem like a mouthful in the first 2 mins, becomes crystal clear minutes later and is consolidated through the examples.
Prof. Leonard, thank you for all of your hard work and dedication. I wear my "Professor Leonard Believes in You" t-shirt with pride.
Amazing. You make it super easy to understand everything
I am really glad to see such confidence in explaining useful courses, beside the tutor is friendly.
Fantastic justification for why we subtract 2 from both +ve and -ve x-intercepts. Trying to teach in this way with a lot of patience is just amazing. Prof. you are simply awesome :))
I didn't know that learning math could be this meditating!
Professor Leonard ,thank for smooth explanation of Descartes Rule of Signs. This is a classical way to determine the number of positive/negative x- intercept of any polynomial.
Thanks for this. It explains this concept in an incredibly easy to understand but concise way. It's not that hard of an idea, but thorough explanations help me retain, and I couldn't find one until now.
THANKU SIR I UNDERSTAND EVERYTHING WHAT YOU THOUGHT
God bless you, professor.
God bless america!!! Beer
U make me fall in love w/ the subject 💓 thanks
Me too!!!!!
Please do some more differential equations topics. Thank you!
This helped me understand, thank you
No one gonna mention that superman is teaching you math
I'm looking into getting my sibling into Professor Leonard for math but I am Canadian so I'm not sure what the order of subjects is for his math lessons. If someone could tell me, it would be very appreciated. In Canada, in gr 12 we start with Advance Funtions into Calculus and Vectors. Then in university we do Calc 1. Of all the math playlist the Professor has, waht order should a young student watch them in? Thanks.
In the United States, it goes Pre-algebra, to Algebra I, to Algebra II, to Precalculus (College Algebra & Trigonometry), to Calculus 1.
I forgot to add in my previous comment that I had learned a lot, so thank you.
I really appreciate your work while I was taking cal 3. Will you do any videos on real analysis?
it's easy to remember seeing as how I watched the last video ~ 20 minutes ago.
So quick question. Whenever we have a cubic function, and we do our real zero test, and then we do Descartes rule of signs, what comes after. So if we have 3 positive real zeros or 1 real zero,
and 2 negative real zeros or 0 negative real zeros,how do we know which ones to test first with synthetic division
Oh haha I saw the explanation video
Thought f of x included all values of x. So how does negative x work? Might have to think on this a while.
thank you :)
Why f(x) for +ve x intercepts and f(-x) for -ve x intercepts ‽
why did you keep your sign for three in the last example????
@0:08 "we have kind of a short video"
- video is 30 mins long
In the last ex at f(x)=3x^6+82x^5+27
We found 2 negative solutions for it , so the remaining 4 are imaginary right ?
Congratulations!! After 30 minutes and 36 seconds you have taught me nothing new!!
Flexing about knowing highschool level math really just shows how much of a loser you are. 🤣
Bruh, what
Hello Sir; thanks so much for the lesson. Please am a level hundred BSC. mathematics student. I will love to become a good mathematician like yourself. Need your guidance please. From Ghana
wtf
@@quant-prep2843 what you mean "wtf"?
-2x^3 - x^2 - 2x -3 has 1 negative intercept. What you said is wrong
No you are wrong... there is no sign change... all negative so ...zero negative x intercept....
someone explain to me why we have to subtract 2 in a much simpler way i didnt get his explanation
Okay, look: Imagine you have a polynomial given to you in a factored form, so say:
f(x)= -2(x-7) * (x+1)^2 * (x+4)^3 * (x^2+9)^2
You can either distribute to find the highest power or degree of your function or do the fake distribution trick he showed us, either way it is going to be x^10. So we could have 10 intercepts. But the factored form of course shows us exactly how many we have: 3. For every factor but for the last one there is a solution. But you notice that the last one does not yield two solutions because:
(x^2+9)^2=0. /square root
x^2+9=0. /-9
x^2=--9 --> We cannot now take square root.
So what we have lost in this factor is the possibility of two additional solutions.
If the expression were x^2-9 we would have had -3 and 3 as additional x intercepts, right?
Every single x raised to an odd power will have an x intercept: x^3 or x^9 or whatever, they have to go through once. So you know you'll not lose any with them. But with even powers there's the possibility of losing either none (they yield real solutions) or strictly two.
When you get the standard form of that function, the x^10 is explicitly given so you know you could have 10. But since you have not neatly factored it and you don't know how your powers will be distributed across the fractions, you do not know which factors will yield solutions and which ones won't: So we have to assume that some of the factors could be such as to be irreducible, and since we don't yet know how many, we account for that by simply always assuming that we'll be losing two solutions. That's why we then have a list of possible x-int, depending on how many factors will turn out to deny solutions by being irreducible.
Hope this helps!
complex solutions means it never touches x axis ... that means we need to take that into account... so subtract all even possibilities out of the counted no of sign change.....
in case of odd I think at least one real solution will be there as Odd function is a journey from +∞ to -∞ or vice versa in North -South Direction ( Y Axis) .. it must pass X axis at least once during its journey via Y axis... where as an even function can totally bypass X intercept by flying high.. something like " complex functions are out of the Radar of X intercept " just plot f(x) =x^2 +36 and see it for yourself...
Because you always get complex roots in pairs (one complex number and other it's conjugate), so if a polynomial is of degree n, then either you get n real roots or you get some complex roots, since complex roots always occur in pairs, there can be either 2 or 4 or 6 of them and that means you get either n-2 real roots, (n-2)-2 or n-4 real roots and so on, that's why we always subtract 2
I am from India I have a gaming channel u should like it ,sorry for bad english
The first one to comment
Are you sure?
@@odie-wankenodie8607 dumb