SOLUTION: 4.a. Use Descartes' Rule of Signs to determine the number of possible positive zeros and the number of possible negative zeros of 12x4 — 67x3 + 108x2 — 47x + 6.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: 4.a. Use Descartes' Rule of Signs to determine the number of possible positive zeros and the number of possible negative zeros of 12x4 — 67x3 + 108x2 — 47x + 6.      Log On


   

Question 113799: 4.a. Use Descartes' Rule of Signs to determine the number of possible positive zeros and the number of possible negative zeros of 12x4 — 67x3 + 108x2 — 47x + 6.
Answer by jim_thompson5910(35256) About Me  (Show Source):
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4 a

Using Descartes' Rule of Signs, we can find the possible number of positive roots (x-intercepts that are positive) and negative roots (x-intercepts that are negative)

First lets find the number of possible positive real roots:

For 12x%5E4-67x%5E3%2B108x%5E2-47x%2B6, simply count the sign changes

Here is the list of sign changes:
  1. 12x%5E4 to -67x%5E3 (positive to negative)
  2. -67x%5E3 to 108x%5E2 (negative to positive)
  3. 108x%5E2 to -47x (positive to negative)
  4. -47x to 6 (negative to positive)



So there are 4 sign changes, this means there are a maximum of 4 positive roots

So the number of positive real roots is 4,2, or 0





Now lets find the number of possible negative real roots

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First we need to find f%28-x%29:

f%28-x%29=12%28-x%29%5E4-67%28-x%29%5E3%2B108%28-x%29%5E2-47%28-x%29%2B6 Plug in -x (just replace every x with -x)

f%28-x%29=12x%5E4%2B67x%5E3%2B108x%5E2%2B47x%2B6 Simplify (note: if the exponent of the given term is odd, simply negate the sign of the term. If the term has an even exponent, then the sign of the term stays the same)

So f%28-x%29=12x%5E4%2B67x%5E3%2B108x%5E2%2B47x%2B6


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Now lets count the sign changes for 12x%5E4%2B67x%5E3%2B108x%5E2%2B47x%2B6:
By looking at 12x%5E4%2B67x%5E3%2B108x%5E2%2B47x%2B6 we can see that there are no sign changes (all the terms are positive).
So there are no negative roots

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Summary:


So there are 4,2, or 0 positive roots and 0 negative roots