Trevor B. answered • 03/29/22
Down-to-earth and here to help
So for some function P(x), the Rational Zero Theorem says that possible zeros of the function can occur at p/q, where p are the factors of the constant a0 (i.e. -6 in this case) and q are the factors of the leading term an (i.e. 2).
The factors of p are ±1, ±2, ±3, and ±6.
The factors of q are ±1 and ±2.
Thus the possible combinations of p/q are ±1/1, ±1/2, ±2/1, ±2/2, ±3/1, ±3/2, ±6/1, ±6/2, which after simplification gives us ±1, ±1/2, ±2, ±3, ±3/2, ±6.
While you can just plug each number into the equation to determine if it's a zero, it is faster to use synthetic division to find a remainder:
1| 2 1 -7 -6
| 2 3 -4
2 3 -4 -10 < this is the remainder, which means that x = 1 would not be a zero of the equation.
-1 | 2 1 -7 -6
| -2 1 6
2 -1 -6 0 < this means that x = -1 is a zero of the equation
If you don't know how to do synthetic division, feel free to look up a guide on the internet. I'm not going to list all the answers, but you should get the idea from this. Simply continue dividing each combination of p/q to see if it comes out as a remainder of 0, or plug each one in to calculate it.
