X^3-3x^2-10x+24 rational zero theorem

The rational zero theorem looks at the number of sign changes in the equation's terms when x is evaluated with a positive or negative number. In your problem, if we insert a positive value for x:
1st term stays +, 2nd term stays  -, 3rd term stays -, last term stays +.
Read from left to right and determine the number of sign changes:
+ - - + [there is a change from + to -, no change from - to -, and a change from - to +] Therefore 2 sign changes. There can be two positive rational zeros, or zero rational zeros [It's always the number of sign changes, and then every other number less than the number of sign changes, and zero]

Let's now do it for a negative x:
1st term becomes -, 2nd term stays -, 3rd term becomes +, last term stays +
- - + + [there is no sign change from - to -, one change from - to +, no change from + to +] Therefore one sign change. There can be one negative rational zero, or zero rational zeros.
This is a difficult concept to type out in a screen box. Please post me a comment if you need additional clarification!

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