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Determine the number of possible positive and negative real zeros for the given function.

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has real coefficients and the constant term is nonzero, so we can apply Descartes’ Rule of Signs.
To determine the number of possible positive real zeros, determine the number of sign changes in .
has sign change(s).
The number of possible positive real zeros is either (Choose one) .

Ask by Watkins Frank. in the United States
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

has 4 sign changes, so there are either 4, 2, or 0 positive real zeros.

Solution

  1. Identify the coefficients in order of descending degree:
    • (from )
    • (from )
    • (from )
    • (from )
    • (from )
    • (constant term)
  2. Count the sign changes:
    • to : no change (both are positive)
    • to : one sign change (positive to negative)
    • to : one sign change (negative to positive)
    • to : one sign change (positive to negative)
    • to : one sign change (negative to positive)
    Total number of sign changes is .
  3. By Descartes’ Rule of Signs, the number of positive real zeros is either equal to the number of sign changes or less than that by an even integer. Thus, the possible numbers of positive real zeros are:
So, the answer is:
has sign change(s), and the number of possible positive real zeros is either , , or .

Answered by UpStudy AI and reviewed by a Professional Tutor

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Simplify this solution

Bonus Knowledge

Let’s dive into the coefficients of . Here’s the breakdown of the signs:
  1. (positive)
  2. (positive)
  3. (negative)
  4. (positive)
  5. (negative)
  6. (positive)
Now let’s count those sign changes:
  • From to : no change
  • From to : change
  • From to : change
  • From to : change
  • From to : change
That gives us a total of four sign changes.
According to Descartes’ Rule of Signs, the number of possible positive real zeros could be equal to the number of sign changes or fewer by an even number. Thus, the possibilities are or or .
Now moving on to the negative zeros, we evaluate :
The signs here for are as follows:
  1. (negative)
  2. (positive)
  3. (positive)
  4. (positive)
  5. (positive)
  6. (positive)
Counting the sign changes:
  • From to : change
  • From to : no change
  • From to : no change
  • From to : no change
  • From to : no change
That’s just one sign change for . Using Descartes’ Rule again, the number of possible negative real zeros is therefore either or .
Summarizing, we have:
  • Positive real zeros: , , or
  • Negative real zeros: or
You can now fill in those final boxes!

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