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 one year ago
Help? Using complete sentences, describe how you would analyze the zeros of the polynomial function f(x) = –3x^5 – 8x^4 +25x^3 – 8x^2 +x – 19 using Descartes’ Rule of Signs.
 one year ago
Help? Using complete sentences, describe how you would analyze the zeros of the polynomial function f(x) = –3x^5 – 8x^4 +25x^3 – 8x^2 +x – 19 using Descartes’ Rule of Signs.

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jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1in f(x), how many sign changes are there?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1notice that going from 8x^4 to +25x^3, there's a sign change from negative to positive

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1do you see this?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1ok there's another from +25x^3 to 8x^2

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1what's another one?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1there's one more

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1from 8x^2 to +x

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1so there are 4 sign changes total in f(x)

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1this means that there are at most 4 positive real roots

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1no that's the same sign change (just in reverse)

P.nut1996
 one year ago
Best ResponseYou've already chosen the best response.04 sign changes=4 positive real roots

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1at most 4 (there could be 0, 1, 2, 3, or 4 positive real roots)

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.14 is the maximum

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1now we must find f(x)

P.nut1996
 one year ago
Best ResponseYou've already chosen the best response.0I'm still really confuzzled...

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1the rule is if f(x) has n sign changes, then there are AT MOST n positive real roots

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1if n = 4, then if f(x) has 4 sign changes, then there are AT MOST 4 positive real roots

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1making more sense?

P.nut1996
 one year ago
Best ResponseYou've already chosen the best response.0I don't know how to formulate an answer from you telling me this...

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1you don't need to find the actual roots you just need to be able to count the possible number and type

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1more specifically, how many positive real roots you could have (in this case, at most 4) you don't need to find the 4 or find out how many actual positive roots there are

P.nut1996
 one year ago
Best ResponseYou've already chosen the best response.0But I need a definite answer, right?

P.nut1996
 one year ago
Best ResponseYou've already chosen the best response.0I can't just say "maybe 4"?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1well that's part of the answer, we still have to find f(x)

P.nut1996
 one year ago
Best ResponseYou've already chosen the best response.0Okay, and how would we go about that ? :)

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1start with f(x) then replace each x with x and simplify

P.nut1996
 one year ago
Best ResponseYou've already chosen the best response.0Could you give me an example?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1start with this f(x) = –3x^5 – 8x^4 +25x^3 – 8x^2 +x – 19 then replace each 'x' with 'x' to get f(x) = –3(x)^5 – 8(x)^4 +25(x)^3 – 8(x)^2 +(x) – 19 then simplify to get ???

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1does that help?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1ok what do you get

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1when you simplify

P.nut1996
 one year ago
Best ResponseYou've already chosen the best response.03x^5=f(x)+x(x(x8x(+25)+8)+1)+19??

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1f(x) = –3(x)^5 – 8(x)^4 +25(x)^3 – 8(x)^2 +(x) – 19 would simplify to f(x) = 3x^5 – 8x^4  25x^3 – 8x^2  x – 19

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1now count the sign changes in f(x) = 3x^5 – 8x^4  25x^3 – 8x^2  x – 19 and tell me how many you got

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.13x^5 is the same as +3x^5

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1so there's only one sign change from +3x^5 to 8x^4

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1so this means that there is at most 1 negative real root

P.nut1996
 one year ago
Best ResponseYou've already chosen the best response.0Oh, I was just looking at it wrong.. Sorry :(

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1So far we found that there are at most 4 positive real roots and at most 1 negative real root

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1so one possibility is that there are 4+1 =5 real roots (4 positive, 1 negative) BUT this is not the only possibility

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1we could have 3 positive real roots and 1 negative real root to have 3+1 = 4 real roots total but the problem with this scenario is that you would only have 51 = 1 complex root...when complex roots should come in pairs

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1so that scenario of 3 positive real roots and 1 negative real root just isn't possible

P.nut1996
 one year ago
Best ResponseYou've already chosen the best response.0Okay So don't worry about the last scenario?

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1anyways, there are a ton of possibilities the good news is that you can sum them all up by saying there are at most 4 positive real roots (you could have 0, 1, 2, 3, or 4 positive real roots) there is at most 1 negative real root (you could have 0 or 1 negative real roots) there are at most 5 complex roots (if there are no real roots at all)

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1in short there are at most 4 positive real roots there is at most 1 negative real root there are at most 5 complex roots

jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.1we can stop there because we don't need to find the actual roots or the counts of the types or roots...just the max of all of the possible types of roots
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