Hahaha ... thank you, Cerebral Assassin. I am glad that you enjoyed the video and found its contents to be 'Crystal Clear.' Warm regards and best wishes to you!
Thank you, Abdul. I suppose digital things 'stay fresh' ^^. I am glad that you found the video to your liking ... even after all these years. Kind regards, and thank you for your enthusiastic comment/compliment!
Thank you, friend. I am glad that you found the video useful. And you are quite observant :-). I have been unable to produce videos for some time (hope/expect to resume next year), but have tried to respond to everyone's comments. Thank you and best wishes!
Thank you, friend. You are right! Once the principles are understood, it is quite an easy process (especially when integers are used). The simplicity of design allows me to introduce viewers to those particular principles in such a way that I hope they are 'crystal clear.' Kind regards to you (and thank you for your feedback).
Thank you, Akash. Things are improving, but we are still addressing some significant challenges. They have delayed my return to creating videos, but I am still hopeful of resuming soon. Kind regards to you!
@@something3476 I am delighted to learn that this video has been of value to you. Thank you very much for taking the time to leave your feedback. It is encouraging and greatly appreciated. Beat wishes for your continuing studies.
@@mariahalqubtiyah5269 Wow! I greatly appreciate your letting me know this ... and it is wonderful to learn that my videos are helping people with their mathematics. Thank you very much Mariah. Please pass on my greetings and congratulations to your classmate, too! Kind regards from Australia.
@@CrystalClearMaths Yes!! my teacher suggested this video to us right after, since we had another quiz this morning. Guess what, I got full marks! I hope you're doing well in life. Thanks again!! Kind regards from Malaysia!
I love the fact that you took the time to leave such an encouraging comment, Denise. Thank you! I am delighted to learn that my explanation of repeated roots was of value to you and, although I cannot control my accent, am glad that it is sufficiently 'easy on your ear' that you enjoy it. Very best wishes to you, and kind regards from Australia.
Because the equation for the x-axis is y = 0. When we set y = 0, we are solving two equations simultaneously ... that of the polynomial (or any function for that matter), and the line y = 0. The results are all the points where the polynomial crosses or touches the x-axis (i.e. the points of intersection). Algebraically, we refer to these values as the 'roots' or 'zeros' of the polynomial. Graphically, they are simply referred to as x-intercepts. Thank you for asking, Tran.
You are very kind, Prithwish. I am grateful for the huge compliment that you have paid me. I am not sure that I deserve it or can live up to it ... but am delighted that you have found my video(s) worth while! Thank you very much for your comment. Kind regards to you ...
Hello ronk. Sorry I have been so slow to reply. I am not sure what arrows you are referring to. If you are asking about the behaviour of the polynomial for large values of x (towards the right end of the x-axis), it has to do with the sign (positive or negative) of the highest power of x. Please clarify for me and I will try to respond. Kind regards to you.
Hi StarSun. If you have a multiple root in a polynomial, for example, y = (x - 5)³, you only have to list the roots as x = 5. If you wish, you could certainly say x = 5 (triple root) ... thereby clarifying that it occurs three times ... but it is normal simply to list the unique roots at the end. Best wishes to you!
Wouldn't y = x^1 (x to the power of 1) just be a straight line, rather than anything curving? If it were an x-value / zero, then would it not just run through the x-axis (like a straight line) rather than turning?
You are quite right about y = x graphs, s2b, but straight lines are just simple members of the family of polynomials. If you think of them as y = (x - 2) for example, or y = 4(x + 1), they will have zeros of x = 2 and x = -1 respectively ... meaning that they pass through the x-axis at those places. Polynomials do not always curve. If they only contain x¹ and xº terms, they will be straight lines. I hope I have answered your questions. If not, please let me know and identify the part of the video that you wish to query. I will certainly help if I can. Best wishes from Australia
I am glad to hear that, Greg. It is always encouraging to learn that these videos make a difference for people. Thank you for letting me know .... and best wishes to you.
You are very welcome, Mahdi ... and I appreciate your taking the time to leave a comment for me. In fact, I like your comment a lot because the 'slogan' for Crystal Clear Mathematics is ... "Easier than you think!" :-) Thank you again, and kind regards.
This will occur when a = 1, Aditya. Then you will have x² + 2x + 1 = (x + 1)², so the repeated root will be x = -1. The way to demonstrate this would be to realise that ax² + 2x + 1 is a quadratic expression. If a quadratic expression is a perfect square (i.e. it has repeated roots, or a double root), the discriminant will be zero. So, Δ = b² - 4ac = (2)² - 4(a)(1) = 4 - 4a = 0. The only solution to this equation is a = 1. So, there is no other possibility. Best wishes for your studies.
I am delighted, Michael! It is always good to hear of people having those 'aha' experiences :-) Thank you very much for letting me know that this video has helped you. Graeme
One of the best maths videos I've ever come across, it made everything Crystal Clear Maths
Hahaha ... thank you, Cerebral Assassin.
I am glad that you enjoyed the video and found its contents to be 'Crystal Clear.'
Warm regards and best wishes to you!
Thank you so much
You are welcome, Sarvarth.
Very nice thanku 👍
You are welcome, friend. :-)
very nice
Thank you, Kumara.
I am glad that you liked the video.
Kind regards to you (and thank you for leaving your comment).
Great vids!
+Gabriel R Thank you, Gabriel.
what a bloody legend you are, 9 years and its still just as good
Thank you, Abdul.
I suppose digital things 'stay fresh' ^^.
I am glad that you found the video to your liking ... even after all these years.
Kind regards, and thank you for your enthusiastic comment/compliment!
Thank you very much, Julia. I am glad that it helped.
I appreciate your taking the time to let me know.
Best wishes,
Graeme
Appreciate the explanation, this makes sooooo much more sense now. And its really cool that you’ve been replying to every comment lol
Thank you, friend.
I am glad that you found the video useful. And you are quite observant :-). I have been unable to produce videos for some time (hope/expect to resume next year), but have tried to respond to everyone's comments.
Thank you and best wishes!
i sorta giggled at how easy you made this. Good on you mate, absolutely amazing tutorial.
Thank you, friend.
You are right! Once the principles are understood, it is quite an easy process (especially when integers are used).
The simplicity of design allows me to introduce viewers to those particular principles in such a way that I hope they are 'crystal clear.'
Kind regards to you (and thank you for your feedback).
Mate you've actual saved me from my maths test tomorrow much appreciated!!!
You are welcome, Zelepher.
I hope your test goes well for you. Please let me know what transpires.
Best wishes to you!
best video i could find! understand it well now, thank you!
I am glad that the concept now makes sense, Iva.
Best wishes for your mathematical (and life) journey.
Loved your video ❤️ . Hope you are doing fine now sir
Thank you, Akash.
Things are improving, but we are still addressing some significant challenges. They have delayed my return to creating videos, but I am still hopeful of resuming soon.
Kind regards to you!
Thank you very much, Justin. I appreciate your feedback.
I try to keep things clear (even if it means repeating things a little).
You are welcome, Elisha.
I am glad that it helped :-)
Fantastic explanation! Thanks
Thank you, very much, Sat NakS. You are welcome.
OMG, so CLEAR!!! thank you SIR!
I searched like hell for this explanation....Great....Loved it :-)
Thank you, Mahaan. I'm glad that you found it and even more delighted that you loved it!
I felt like I was getting taught by King Pin from Daredevil. Thank you!
Hahaha ... thank you, Ulises :-)
thank you for this video, I was going through my notes from class but for some reason "bouncing off" didn't make sense. But this video helped a lot.
@@something3476 I am delighted to learn that this video has been of value to you. Thank you very much for taking the time to leave your feedback. It is encouraging and greatly appreciated. Beat wishes for your continuing studies.
You helped my classmate ace our quiz this morning!!! Thank you
my classmate too!!!
@@fatinnabilah2577 omg I can't believe i met u here 💗💗💗
@@mariahalqubtiyah5269 Wow! I greatly appreciate your letting me know this ... and it is wonderful to learn that my videos are helping people with their mathematics.
Thank you very much Mariah. Please pass on my greetings and congratulations to your classmate, too!
Kind regards from Australia.
@@fatinnabilah2577 Greetings to you from Australia, Fatin. I hope you find useful videos here. :-)
@@CrystalClearMaths Yes!! my teacher suggested this video to us right after, since we had another quiz this morning. Guess what, I got full marks! I hope you're doing well in life. Thanks again!!
Kind regards from Malaysia!
Why are u ao slow.
Probably ... because ... I ... am ... simply ... getting ... old ... :-)
@@CrystalClearMaths yes u explained very well sir .and u r not getting old probably more younger than before .thanks for ur help sir..
@@pratikkumarjha5995 Hahaha ... thank you, kindly, Pratik. I'm glad I could help.
Best wishes to you!
I love your explaination, I love your accent, I just love you thanks for this video
I love the fact that you took the time to leave such an encouraging comment, Denise. Thank you!
I am delighted to learn that my explanation of repeated roots was of value to you and, although I cannot control my accent, am glad that it is sufficiently 'easy on your ear' that you enjoy it.
Very best wishes to you, and kind regards from Australia.
Why must the expression equal 0
Because the equation for the x-axis is y = 0.
When we set y = 0, we are solving two equations simultaneously ... that of the polynomial (or any function for that matter), and the line y = 0. The results are all the points where the polynomial crosses or touches the x-axis (i.e. the points of intersection).
Algebraically, we refer to these values as the 'roots' or 'zeros' of the polynomial. Graphically, they are simply referred to as x-intercepts.
Thank you for asking, Tran.
Sir is the definition of not all heroes wear capes
You are very kind, Prithwish.
I am grateful for the huge compliment that you have paid me. I am not sure that I deserve it or can live up to it ... but am delighted that you have found my video(s) worth while!
Thank you very much for your comment.
Kind regards to you ...
how do you know when to draw the arrows on the left or right.
Hello ronk. Sorry I have been so slow to reply.
I am not sure what arrows you are referring to.
If you are asking about the behaviour of the polynomial for large values of x (towards the right end of the x-axis), it has to do with the sign (positive or negative) of the highest power of x. Please clarify for me and I will try to respond.
Kind regards to you.
When listing the factors and roots, do you list that repeating factor and root once or you list it based on how many times it repeats?
Hi StarSun.
If you have a multiple root in a polynomial, for example, y = (x - 5)³, you only have to list the roots as x = 5.
If you wish, you could certainly say x = 5 (triple root) ... thereby clarifying that it occurs three times ... but it is normal simply to list the unique roots at the end.
Best wishes to you!
A big thank you. The video was of immense help.
Wouldn't y = x^1 (x to the power of 1) just be a straight line, rather than anything curving? If it were an x-value / zero, then would it not just run through the x-axis (like a straight line) rather than turning?
You are quite right about y = x graphs, s2b, but straight lines are just simple members of the family of polynomials. If you think of them as y = (x - 2) for example, or y = 4(x + 1), they will have zeros of x = 2 and x = -1 respectively ... meaning that they pass through the x-axis at those places. Polynomials do not always curve. If they only contain x¹ and xº terms, they will be straight lines.
I hope I have answered your questions. If not, please let me know and identify the part of the video that you wish to query. I will certainly help if I can.
Best wishes from Australia
awesomely explained
Thank you, xoxoxoxoxoxoxo. I am glad that you enjoyed the video.
Your comment is most encouraging.
thank you Tom Hanks :)
It is a bit of a 'stretch,' but thank you, i'm diene.
I am glad that you enjoyed the video.
very help full videos
I am glad that you found them useful, Vivek. Thank you.
really helpful !
Thank you for your feedback, Harsimran. I am glad that my video was a help to you.
Very well explained. I like your style.
Thank You. Great Video on multiplicity.
You are very welcome, Avis. I am glad that you liked it and found it helpful!
So so helpful. Thank you.
This was very helpful for my math grade
I am glad to hear that, Greg. It is always encouraging to learn that these videos make a difference for people.
Thank you for letting me know .... and best wishes to you.
Helpful
You are welcome, Vishnu. I am glad that you liked the video.
Kind regards to you.
Thanks a lot you made it easier than I thought it was
You are very welcome, Mahdi ... and I appreciate your taking the time to leave a comment for me.
In fact, I like your comment a lot because the 'slogan' for Crystal Clear Mathematics is ... "Easier than you think!" :-)
Thank you again, and kind regards.
Concise and comprehensive. Much appreciated sir!
Thank you very much, Crinnack.
I try not to waste your time :-)
Best wishes to you.
Sir when will ax2+2x+1 have repeated roots when
1)a=0
2)a=1
3)a=-1
4)a=2
And also why for your answer sirrrrr please reply me.
This will occur when a = 1, Aditya.
Then you will have x² + 2x + 1 = (x + 1)², so the repeated root will be x = -1.
The way to demonstrate this would be to realise that ax² + 2x + 1 is a quadratic expression.
If a quadratic expression is a perfect square (i.e. it has repeated roots, or a double root), the discriminant will be zero.
So, Δ = b² - 4ac = (2)² - 4(a)(1) = 4 - 4a = 0. The only solution to this equation is a = 1. So, there is no other possibility.
Best wishes for your studies.
@@CrystalClearMaths very much thank u sir. You are a genius teacher wishing that god will help you and keep you safe.
Sir can you please give me your contact /whatsapp no.
@@user-Danger696 You are welcome, Aditya (and very kind).
May God bless you, too!
Once again thank u sir
Well, that cleared up a fundamental issue I had with doing these problems!
I am delighted, Michael!
It is always good to hear of people having those 'aha' experiences :-)
Thank you very much for letting me know that this video has helped you.
Graeme
Thank You.
It is actually Crystal Clear!!!!
+Love Math I am glad, friend :-)
Thank you for letting me know.
Best wishes for your studies and your love of mathematics.
Thank you, this really helped me :)
You are welcome, David. It encourages me to know that my efforts have helped you.
Best wishes,
Graeme
A heartfelt thank you
You are welcome, Shifa. I am glad that I could help you.
really good explanation.
Thank you!
clear and easy to learn.
Thank you, Cute DIYs. That is very encouraging feedback. I am glad that my video helped you.
Thank you so much sir.
You are very welcome, Aswin.
Thank you for letting me know that this video helped you.
Warm regards,
Graeme
💕💕💕
:-)
Thank you, Simple Craftz.
Thank you! This video really helped me.
That is wonderful, Arushi.
Thank you for letting me know, and best wishes for your studies.