The Rational Zero Theorem to find potential zeros

[lolily]

My textbook has an example that says this: "Use the rational zero theorem to list all the potential rational zeros. D(x)= 2x^4 -7x^3 +x^2+7x -3." The next step is to list the factors of a₀ and a₃, which are ±{1,3} and ±{1,2}. The textbook goes straight from this to saying that the answer to the problem is ±{1/2,1,3/2,3}.

I don't understand where the fractions in the answer come from, and why 2 is not listed in the answer. Thank you in advance to anybody who can explain it to me

 

[Dr.Peterson]

The potential rational zeros are fractions whose numerator is one of {1, 3}, and whose denominator is one of {1, 2}. These are:

• 1/1
• 1/2
• 3/1
• 3/2

Why do you think 2 should be an answer? Are you swapping the sets?

 

[lolily]

No, I understand now. For some reason, I thought that all of the potential zeros where just the combination of the factors of a0 and a3. My textbook didn't explain it very well, but now I get it. Thank you.

 

[Dr.Peterson]

Good. The reason I asked was that it is common for students at first to get the order wrong. For them, I suggested remembering that the solution to the very simple equation ax + b = 0 is x = -b/a, where the constant term is on top, and the leading coefficient is on the bottom, just as in the RZT.

 

[Steven G]

I look at the equation 1x-2=0 whose solution is x=2. Then I think do I put the factors of 2 for the numerator or the denominator. Obviously I put the factors of 2 in the numerator and the factors of 1 in the denominator as that is the only way to get 2 as a possible root.

EDIT: after looking at Dr Peterson example of ax+b=0 a 2nd time I think that is the way to go in deciding which order to use!