## LovinBachata Group Title Find The Four Real Zeros Of The Polynomial. f(x)= x^4+6x^3-15x^2-152x-240 one year ago one year ago

1. mathmate Group Title

Descartes rule of signs tells us that there is only one positive root. Factoring theorem tells us that the rational roots would be factors 240 (which has many). To guess the positive root, seeing that the first two terms are positive, with small coefficients, and the remaining (negative) have large coefficients, we can try large factors, such as 4 (4^4=256), or 5 (5^4=625) by evaluating f(4) or f(5). Once we find the first one, the rest are negative roots of the remaining cubic.

2. LovinBachata Group Title

so the anser would be -4,-4,-3,5 ?

3. mathmate Group Title

Yes, how did you find them?

4. LovinBachata Group Title

i first used synthetic division using the number 4 and -4 and kept dividing until i had four solutions

5. mathmate Group Title

Yep, that's great! Not everyone is familiar with synthetic division. How did you get to start with 4 and -4?

6. LovinBachata Group Title

Synthetic division is easier for me to use and because you gave me those numbers

7. mathmate Group Title

I hope you'd be able to find those starter numbers in an exam using Descartes rule of signs and the factor theorem!

8. LovinBachata Group Title

I Hope So Tooo! Thank You Soo Much

9. mathmate Group Title

You synthetic division capability is a great asset in solving equations. Another way is to evaluate f(x) by nesting, for example, for f(5), you can almost do it by inspection (mentally): f(5)=(((5+6)*5-15)*5-152)*5-240 =((55-15)*5-152)*5-240 =(200-152)*5-240 =48*5-240 =0

10. LovinBachata Group Title

11. mathmate Group Title

Assuming you have a starter number of 5, then evaluating using nesting (as shown above) is fast, accurate and less error prone... with a little practice. The starter number remain guesses, with he help of Descartes Theorem and Factor theorem. Are you already familiar with these two?

12. LovinBachata Group Title

im only familiar with the factor theorem but i dont remember Descartes theroem do you usuallly look ath the last numbeer?

13. mathmate Group Title

If the leading coefficient is 1, then only the last number counts. If the leading coefficient id different from one, then the possible factors are of the form (ax+b) where a is a factor of the leading coefficient, and b is a factor of the last number (constant term). Descartes rule tells us that the maximum number of positive roots is equal to the number of sign changes. In the above case, sign only changed once from + to -, so there is only one positive root, and a maximum of three negative roots. Maximum means 2,4,6... can be replaced by complex roots, since complex roots always come in pairs. For example, in our case, there is one positive root (cannot be replaced by two complex), and one or three real negative roots. Since the question says 4 real roots, we can count on 3 negative (real) roots.

14. mathmate Group Title

* replace (ax+b) by $$(ax\pm b)$$

15. LovinBachata Group Title

so the fact that there is one postive sign means that the solutions would have one positive solution?

16. mathmate Group Title

It's the $$change$$ in sign that counts. x^2-2x+1=0 has two positive roots, but x^2-x-2=0 has only one.