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Polynomial Inequalities

If I have the critical values for the cubic equation y=ax3+bx2+cx+d and am given the inequality ax3+bx2+cx+d>z, or ax3+bx2+cx+d<z, how should one go about finding the exact form of the inequality for x? (We can assume the solutions are known.) I don't want to draw graphs to do this.

Separately, if roots are repeated (e.g. I have a quadratic inequality (x-8)2<0), can I simply take the square root and then solve like a linear inequality (to get x<8)?

Thanks.
Original post by Big-Daddy
I don't want to draw graphs to do this.


Then sketch not draw. Why wouldn't you want to answer the question properly?
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Big-Daddy
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Original post by Mr M
Then sketch not draw. Why wouldn't you want to answer the question properly?


There must be a reasonably simple way of doing it which is more mathematically rigorous than sketching?
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TenOfThem
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Original post by Big-Daddy
There must be a reasonably simple way of doing it which is more mathematically rigorous than sketching?


find values of f(x) either side of the critical values
Original post by Big-Daddy
There must be a reasonably simple way of doing it which is more mathematically rigorous than sketching?


Sketching can be the most efficient form of mathematical communication. About 7 years ago I attended the STEP course for teachers and they urged us to encourage students to solve cubic inequalities by sketching rather than by testing various points / making a table etc.
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davros
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Original post by Big-Daddy

Separately, if roots are repeated (e.g. I have a quadratic inequality (x-8)2<0), can I simply take the square root and then solve like a linear inequality (to get x<8)?

Thanks.


I think you'll find that (x-8)^2 < 0 doesn't have any real solutions anyway :smile:
Reply 6
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Big-Daddy
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13
Original post by Mr M
Sketching can be the most efficient form of mathematical communication. About 7 years ago I attended the STEP course for teachers and they urged us to encourage students to solve cubic inequalities by sketching rather than by testing various points / making a table etc.


Hmm ok, I will go ahead with what you suggest.

Would they ever ask you to solve a quartic or higher degree inequality at A-level? The reason I ask is because carefully drawing (even sketching) the curves for higher degree polynomials will start to take time.
Reply 7
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TenOfThem
18
Original post by Big-Daddy
Hmm ok, I will go ahead with what you suggest.

Would they ever ask you to solve a quartic or higher degree inequality at A-level? The reason I ask is because carefully drawing (even sketching) the curves for higher degree polynomials will start to take time.


why do you think that sketching takes time?
Original post by Big-Daddy
Hmm ok, I will go ahead with what you suggest.

Would they ever ask you to solve a quartic or higher degree inequality at A-level? The reason I ask is because carefully drawing (even sketching) the curves for higher degree polynomials will start to take time.


I shouldn't think so but this shouldn't take any time at all. You said you already know the critical points so you should be able to produce the sketch within about 5 seconds.

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