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How does one solve a polynomial absolute value inequality? say, |(x-a)(x-b)(x-c)|>d

Mathematics
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No joke, I got a prob on a quiz for my AP BC calc class and I miserably got it wrong...
the lesson mentioned inequalities for polynomials, but not polynomial abs value probs.
And, if I'm supposed to use a graphing calculator, tell me, because I honestly have no idea.

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Other answers:

  • phi
I think these are tricky. I would re-write as (x-a)(x-b)(x-c) - d > 0 or -(x-a)(x-b)(x-c) - d > 0 multiply out and then factor to find the zeros then look for intervals where the product of the factors are positive very messy...
Yes, but linear absolute value problems can have extraneous solutions...
  • phi
then check the possible answers in the original equation
So I probs just want a graphing calculator?
  • phi
off hand, I don't know. do you have a specific example?
Let me access my course. Sec
|x^3-x^2+x-1|<3 "Round your answer to three decimal places and place in ascending order with a comma in between." Answer: -0.811
yeah, and my answer was from wolfram. I have no idea what I was supposed to do. and the lesson was not helpful at all. Didn't even mention it.
  • phi
I'll think about it, as this is not something I've looked at very carefully. But I not that x=1 gives 0 so 1 is part of the solution space. Maybe by tomorrow I'll have something to post.
Alright. Thanks for looking at it with me.
if you graph the inner polynomial, whenever it goes negative, it just mirrors above the x axis
well, the curve does, then pick a point that is clearly not on the curve to test
|dw:1348175550151:dw|
|dw:1348175643801:dw|
o.o
it's a fake 6 degree polynomial!
|dw:1348176111960:dw|
Mmm.
its just the normal 3 degree with its negative parts popped up; since the absolute value of a number is never negative.
Yeah. and then we draw that line. any way of an analytic solution, or are we just going to have to graph this guy?
im curious why the y=3 line
i see it now
yes, that y=3 line represents where the x axis would be if we subtract 3 from each side
|x^3-x^2+x-1| < 3 consider the values such that (x^3-x^2+x-1) = 3 -(x^3-x^2+x-1) = 3
x^3-x^2+x-4 = 0 and x^3-x^2+x+2 = 0 there is no good way to do a general cubic solution is there
yeah I'm not cardano, I have 30 minutes to do this xD. I think it was a calculator prblem lol
do you need exact or approx ?
approx.
x = 1.743 and -0.811 is what the google gives me as approx zeros
we can test those in the original format
kay
at x=1.743; |x^3-x^2+x-1| -3 = 0.000xxx at x=-0.811; |x^3-x^2+x-1| -3 = 0.00xxxx and of course, when x=0, 1-3 is less than 0 so it should be an interval like: (-0.811,1.743) or however exact you need it to be
I see. So that's what I'm supposed to do. Thank you.
http://www.wolframalpha.com/input/?i=%7Cx%5E3-x%5E2%2Bx-1%7C+-3%3C0 it even has the interval as a number line at the bottom
  • phi
Here is what I came up with |x^3-x^2+x-1|<3 we can do this with a graphing calculator Case 1: x^3-x^2+x-1 < 3 x^3-x^2+x-4 < 0 Examine the roots of x^3-x^2+x-4 = 0 Find 1 real root at x= 1.743 Graph of the function shows f(x)<0 for x < 1.743 Case 2: -(x^3-x^2+x-1) < 3 or, negating and flipping the sense of the relational operator x^3-x^2+x-1 > -3 x^3-x^2+x+2 > 0 Find the roots of f(x)= x^3-x^2+x+2 =0 1 real root at x= -0.811 Graph shows f(x) > 0 for x > -0.811 Intersection of both regions x < 1.743 x > -0.811 Gives -0.811 < x < 1.743
I get why wolfram gave me the wrong answer. Cause it split it up into two inequalities. So mad lol. and @phi , thanks for the answer :D
  • phi
Sounds like you are ready for these... give 'em hell
Too late :o I got into BC calc 2.5 weeks late and I'm doing catch up work. that right there was week 1 or something, and I'm at week 2.5 tomorrow ;) But thankfully I understand the concept, so I can take it on in event of unit test.
It's not even calculus and I got it wrong. xS

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