I'm HOPING this is the last problem I need help on.. :-/
List the possible rational zeros of f using the rational zero theorem.
f(x) = 2x^3 - x^2 + 5x + 6
I applied the rational zero theorem, but none of the factors worked :(
You are right, it does not factor over the rational numbers.
Try this program
http://www.1728.com/cubic.htm
it shows one real solution, and two imaginary roots.
To find the possible rational zeros of a polynomial using the rational zero theorem, you need to consider all the possible rational numbers that can be obtained by dividing a factor of the constant term by a factor of the leading coefficient.
In the given polynomial f(x) = 2x^3 - x^2 + 5x + 6, the constant term is 6, and the leading coefficient is 2.
To find the factors of 6, we can consider ±1, ±2, ±3, and ±6. To find the factors of 2, we can consider ±1 and ±2.
Now we need to form possible rational zeros by dividing a factor of the constant term by a factor of the leading coefficient.
Possible rational zeros can be obtained by dividing ±1, ±2, ±3, and ±6 by ±1 and ±2.
So, the possible rational zeros are:
±1/1, ±1/2, ±2/1, ±2/2, ±3/1, ±3/2, ±6/1, ±6/2.
Simplified, these become:
±1, ±1/2, ±2, ±3/2, ±3, ±6.
Now, it is important to note that while the rational zero theorem gives you the possible rational zeros, it does not guarantee that any of these numbers are the actual zeros for the polynomial. The polynomial may have irrational or complex roots as well.