Understanding Polynomial and Rational Functions: Tips and Tricks

[NickK]

Right now we are Studying Polynomial and Rational Functions and some things just have me puzzled. Such as

1. Is the degree of the polynomial the biggest exponent in the polynomial of x?
2. Whats the best way (or any way) to find the wxact zeros of this = 32x^4-28x^3+113x^2-112x-60?
3. How do you write the equation above as the product of linear and quadratic factors that are irreductible over the reals?


Any help would be appreciated

 

[shmoe]

1. Is the degree of the polynomial the biggest exponent in the polynomial of x?

Yes. For example, the beast you give below has degree 4.

2. Whats the best way (or any way) to find the wxact zeros of this = 32x^4-28x^3+113x^2-112x-60?

Hrm, that's a biggie. Start with the rational root theorem that states if p/q is a rational root of this polynomial then p is a factor of 60 and q is a factor of 32. Have you seen this before? Try all these candidates as roots until you find one. Then divide out by the corresponding linear factor and repeat the process. One thing to notice to save a bit of time is that if q=1, then if p is odd and p/q, which just equals p, is a root you'd have:

32p^4-28p^3+113p^2-112p-60=0

or

even#-even#+odd#-even#-even#=even#

Which is impossible. So if q=1, p must be even. This means you don't have to check plus or minus 1/1, 3/1, 5/1, or 15/1 as candidates for roots.

3. How do you write the equation above as the product of linear and quadratic factors that are irreductible over the reals?

Try the above first, and report back what you've got. So you don't despair, there are indeed rational roots for this guy.

 

[NickK]

thanks a lot. I got it now