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inequalities involving cubic equations

x^2+7x is greater than or equal to 6+2x^3. I can solve this using the factor theorem, but this is a C1 question and the factor theorem isn't supposed to be known at this stage. Is there another way of solving this? Any pointers greatly appreciated.

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x^2+7x is greater than 6+2x^3

I can solve this using the factor theorem, but this is a C1 question and you are not supposed to know that at this stage. Is there another way of solving this? Any pointers greatly appreciated.
Reply 2
Avatar for Notnek
Notnek
21
Original post by chatterclaw73
x^2+7x is greater than 6+2x^3

I can solve this using the factor theorem, but this is a C1 question and you are not supposed to know that at this stage. Is there another way of solving this? Any pointers greatly appreciated.

To solve this algebraically (as opposed to graphically) you need the roots of the cubic which requires methods beyond C1. Where did you get this question from and have you posted the whole question? Can you please post a picture of it?
(edited 5 years ago)
I am a self learner working from ocr syllabus. The question is from a CD ROM which came with the book. I know about the roots of the cubic eqution from this book. I have three equations as follows. (I will substitute a for alpha, b for beta and c for gamma)
a+b+c=0.5
ab+ac+bc=-3.5
abc=-3
I get a bit lost with the algebra then.
p.s The CD is an extension designed to test the more able students.

Original post by Notnek
To solve this you need the roots of the cubic which requires methods beyond C1. Where did you get this question from and have you posted the whole question? Can you please post a picture of it?
Reply 4
Avatar for Notnek
Notnek
21
Original post by chatterclaw73
I am a self learner working from ocr syllabus. The question is from a CD ROM which came with the book. I know about the roots of the cubic eqution from this book. I have three equations as follows. (I will substitute a for alpha, b for beta and c for gamma)
a+b+c=0.5
ab+ac+bc=-3.5
abc=-3
I get a bit lost with the algebra then.
p.s The CD is an extension designed to test the more able students.

Can you please tell us what the level of this book is? Is it C1? Further Maths?

Also, please post the question exactly as you're seeing it (a picture would be helpful). After your two posts it's not clear what the actual question was.
Reply 5
Avatar for mqb2766
mqb2766
21
Original post by chatterclaw73
x^2+7x is greater than or equal to 6+2x^3. I can solve this using the factor theorem, but this is a C1 question and the factor theorem isn't supposed to be known at this stage. Is there another way of solving this? Any pointers greatly appreciated.


The obvious way is to factor the cubic and one of the roots is obvious (factor theorem), just leaving a quadratic for the other two. Sounds like what you have done though.
Which board are you doing / where did the question come from?
you can see by inspection that one root is 1. so there is a factor of ( x - 1 )...then you can easily find the other factors.
Original post by mqb2766
The obvious way is to factor the cubic and one of the roots is obvious (factor theorem), just leaving a quadratic for the other two. Sounds like what you have done though.
Which board are you doing / where did the question come from?


The book is titled AS CORE FOR OCR(Rosemary Emmanuel/John Wood). The question is from an extension exercise on a CD-ROM which came with the book. I think it may have something to do with the roots of the cubic equation and I have three equations. I will substitute a for alpha, b for beta and c for gamma.
a+b+c=0.5
ab+ac+bc=-3.5
abc=-3
The algebra gets a bit tricky for me then.
Reply 8
Avatar for mqb2766
mqb2766
21
Original post by chatterclaw73
The book is titled AS CORE FOR OCR(Rosemary Emmanuel/John Wood). The question is from an extension exercise on a CD-ROM which came with the book. I think it may have something to do with the roots of the cubic equation and I have three equations. I will substitute a for alpha, b for beta and c for gamma.
a+b+c=0.5
ab+ac+bc=-3.5
abc=-3
The algebra gets a bit tricky for me then.


Is it
https://www.amazon.co.uk/Core-Maths-Longman-Advanced-CD-ROM/dp/0582842352
If so, what year is it?
Reply 9
Avatar for Notnek
Notnek
21
Original post by chatterclaw73
The book is titled AS CORE FOR OCR(Rosemary Emmanuel/John Wood). The question is from an extension exercise on a CD-ROM which came with the book. I think it may have something to do with the roots of the cubic equation and I have three equations. I will substitute a for alpha, b for beta and c for gamma.
a+b+c=0.5
ab+ac+bc=-3.5
abc=-3
The algebra gets a bit tricky for me then.

But you mentioned inequalities earlier. I'm still unsure about what the whole question actually is.

Also, I hope you are using a new spec textbook in addition to this one which is very old and not designed for the new spec.
Original post by mqb2766

The year of publication is 2004. Not sure about the code but it is very similar to the book code.
Original post by chatterclaw73
The year of publication is 2004. Not sure about the code but it is very similar to the book code.


Can't say for definite, but the syllabus has probably changed a bit. I'd make sure you got the current books, esp. if you're self taught.
For this problem, x = 1 is an obvious root, then you factor 2x^2+x-6 to get the other two at 1.5 and -2. The obvious root does involve the factor theorem (slightly), but that's the obvious way to do the problem.
Original post by Notnek
But you mentioned inequalities earlier. I'm still unsure about what the whole question actually is.

Also, I hope you are using a new spec textbook in addition to this one which is very old and not designed for the new spec.


The whole question is solve x^2+7x is greater than 6+2x^3.
I am not really interested about specs as I learn for my own pleasure. I have all sorts of academic books from way back!
Original post by mqb2766
Can't say for definite, but the syllabus has probably changed a bit. I'd make sure you got the current books, esp. if you're self taught.
For this problem, x = 1 is an obvious root, then you factor 2x^2+x-6 to get the other two at 1.5 and -2. The obvious root does involve the factor theorem (slightly), but that's the obvious way to do the problem.


Ok. thanks. I am self taught, but I do it for pleasure, not to achieve any particular standard. I think I am probably quite advanced in some areas and lacking in others.
Reply 14
Avatar for Notnek
Notnek
21
Original post by chatterclaw73
The whole question is solve x^2+7x is greater than 6+2x^3.
I am not really interested about specs as I learn for my own pleasure. I have all sorts of academic books from way back!

Ah okay so I assume you won't be doing the A Level maths exams? Can you please tell me the page number where the question is located? I don't know why you posted these equations...

a+b+c=0.5
ab+ac+bc=-3.5
abc=-3
I compared the coefficients of the general cubic to the one formed in the question. The question is on the CD not the book. I passed C1 and C2 and M1 in 2009, but no I am not working towards any exam at the minute.
Original post by Notnek
Ah okay so I assume you won't be doing the A Level maths exams? Can you please tell me the page number where the question is located? I don't know why you posted these equations...

a+b+c=0.5
ab+ac+bc=-3.5
abc=-3
Reply 16
Avatar for Notnek
Notnek
21
Original post by chatterclaw73
I compared the coefficients of the general cubic to the one formed in the question. The question is on the CD not the book. I passed C1 and C2 and M1 in 2009, but no I am not working towards any exam at the minute.

Ah I see now what you're doing - you're trying to use further maths "roots of a polynomial" methods to solve this. These methods won't help here as you've found out yourself :smile:

If this is an AS book then it will include the factor theorem so I assume you're meant to solve it using that?
Reply 17
Avatar for Demee
Demee
2
You can substitute in a value of x :smile:
Original post by chatterclaw73
x^2+7x is greater than or equal to 6+2x^3. I can solve this using the factor theorem, but this is a C1 question and the factor theorem isn't supposed to be known at this stage. Is there another way of solving this? Any pointers greatly appreciated.
Judging by the responses I have had to this problem, it does seem as if the factor theorem is the way to go, but although I know the theorem, It isn't introduced until much later in the book I am working on and so i assumed there was an easier method. Thank you for taking the time to advise and help. I am sure it won't be long before i am annoying you again! Take care.
Original post by Notnek
Ah I see now what you're doing - you're trying to use further maths "roots of a polynomial" methods to solve this. These methods won't help here as you've found out yourself :smile:

If this is an AS book then it will include the factor theorem so I assume you're meant to solve it using that?
Original post by Demee
You can substitute in a value of x :smile:


Yes I now have the answer. Thank you for responding anyway.

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