SOLUTION: I need help with the following problem. Suppose f(x)=5x^3-4x^2-5x-35 a) list all possible rational zeros for the polynomial b) Use synthetic division to find at least 1 ratio

Click here to see ALL problems on Polynomials-and-rational-expressions

Question 193568: I need help with the following problem.
Suppose f(x)=5x^3-4x^2-5x-35
a) list all possible rational zeros for the polynomial
b) Use synthetic division to find at least 1 rational zero.
For the possible rational zeros I found: +/- 1, +/- 1/5, +/- 5, +/- 7, +/- 7/5, +/- 35. Is there one that I am missing? Because when I graphed it on the calculator there was a zero at 12/5 or 2.4 and none of my zeros equal that.
Please help me at your earliest convenience
thanks

Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
You have to be careful when it comes to using the graphing calculator to find the solutions. It turns out that 2.414679875... is a solution (not 2.4). So is NOT a solution.

a)

Any rational zero can be found through this equation

where p and q are the factors of the last and first coefficients

So let's list the factors of 35 (the last coefficient):

Now let's list the factors of 5 (the first coefficient):

Now let's divide each factor of the last coefficient by each factor of the first coefficient

Now simplify

These are all the distinct rational zeros of the function that could occur

which can be sorted and rewritten as

So you have the correct possible rational roots.

b)

From here, simply perform synthetic division using all of the possible zeros to determine which possible roots are actual roots.