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P.nut1996
 2 years 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.
P.nut1996
 2 years 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
 2 years ago
Best ResponseYou've already chosen the best response.1in f(x), how many sign changes are there?

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

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

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

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

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

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

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

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

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

jim_thompson5910
 2 years 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
 2 years ago
Best ResponseYou've already chosen the best response.14 is the maximum

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

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

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

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

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

jim_thompson5910
 2 years 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
 2 years ago
Best ResponseYou've already chosen the best response.0Okay, and how would we go about that ? :)

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

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

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

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

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

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

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

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

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

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

jim_thompson5910
 2 years 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
 2 years 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
 2 years 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
 2 years 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
 2 years ago
Best ResponseYou've already chosen the best response.0Okay So don't worry about the last scenario?

jim_thompson5910
 2 years 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
 2 years 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
 2 years 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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