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# Polynomial Inequalities Watch

1. 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.
I don't want to draw graphs to do this.
Then sketch not draw. Why wouldn't you want to answer the question properly?
3. (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?
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
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.
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
7. (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.
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?
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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Updated: March 26, 2013
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