## HelpMe94 3 years ago Use the Descartes's Rule of Signs to determine the possible numbers of positive and negative zeros of the function: f(x)= -5x^3+X^2-X+5

1. mathmate

How many times did the expression changed sign?

2. HelpMe94

3

3. HelpMe94

i think

4. mathmate

There is then a MAXIMUM of 3 positive roots, but it could also be 1.

5. HelpMe94

why 1?

6. mathmate

Now flip the sign of the coefficients of the odd powers. How many times does the new expression change sign?

7. HelpMe94

Is that 5X^3+X^2+X+5

8. mathmate

The number of sign changes is the maximum number of positive roots, including complex. If there are two complex roots (as is the case), then there is only one (real) positive root.

9. HelpMe94

ok

10. HelpMe94

so the negative right converted right?

11. mathmate

Yes, the number of sign changes of the new expression (i.e. replace x by -x) gives the MAXIMUM number of negative roots.

12. HelpMe94

So for this one there is none for the negative and 1 for the positive

13. HelpMe94

so how do you know there is one and not 3 for the positive?

14. mathmate

Using the Descartes rule of signs, we do know that there is no negative root. But we do not know if there is ONE or THREE positive roots.

15. HelpMe94

ok so the answer for positive is 1 and 3 and negative is 0

16. mathmate

You can find that out by factoring the expression. A cubic has at least one real root, and we know that it is positive in this case.

17. mathmate

correct.

18. HelpMe94

one other question:

19. HelpMe94

Use the Descartes's Rule of Signs to determine the possible numbers of positive and negative zeros of the function: f(x)=3X^2+2X^2+x+3

20. HelpMe94

how does this work then?

21. HelpMe94

the positive looks like there is none and same with the negative?

22. mathmate

Now, let's see. (1) The expression as is has no change in sign, so what do you conclude? (2) flip the sign of the odd powers to give -3X^3+2X^2-x+3, sign changed 3 times, what do you conclude?

23. HelpMe94

wait why is 3X^2 flipped?

24. mathmate

I supposed you have a typo! It's 3x^3, ... or not?

25. HelpMe94

let me see...

26. HelpMe94

27. HelpMe94

Use the Descartes's Rule of Signs to determine the possible numbers of positive and negative zeros of the function: f(x)=3X^3+2X^2+x+3

28. HelpMe94

ok so the positive is none then the negative is -3X^3+3X^2-X+3, so 2 and 1

29. mathmate

No positive is correct. Can you repeat the conclusion for negative?

30. HelpMe94

i don't understand

31. HelpMe94

of so negative is 3 and 1

32. mathmate

The number of possible roots always differ by an even number because complex roots, if any, come in pairs. 1 and 2 will not be correct. 1 and 3 would be.

33. HelpMe94

oh ok

34. mathmate

Cool!

35. HelpMe94

cool?

36. mathmate

Perhaps more exactly, 1 OR 3.

37. HelpMe94

ok thanks that makes sense

38. mathmate

You're welcome! :)