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## Algebra II with Trig Ch4.2

Question from Rachel:

I’m on 4.2 of Algebra 2 at the moment.  I made it through the lesson and understood it until I got to the last section, titled “Finding Rational and Imaginary Zeros.” I understood most of that with the exception of a couple of things. Here are my questions.

#1) Why, when using synthetic division to find the zeros of the polynomial does the book only use the positive version of the possible numbers? For example, +/-1 and +/-5 were the possible rational zeros in Example 5. In the division table, they only used a positive 1. I know that was because they found the zero right off, but from what they’ve been doing, they would only have used positive 1 and positive 5. Could you tell me why?

#2) In the matched problem after example 5, the polynomial looks like this:
P(x) = X^4 + 4X^3 + 10X^2 + 12X + 5.

When you find the possible rational zeros (+/- 1 and +/- 5), and plug them into the synthetic division table, none of them (as positive numbers – like the book has been using) turn out to be zeros!
On the 5 for example, 5 times 1 is 5. Added to 4, it becomes 9. 9 times 5 is 40. Added to 10 it becomes 50. 5 times 50 is 250. Added to 12, it becomes 262. 5 times 262 is 1310. Added to 5, it is 1315.

Now I know that that’s not supposed to be happening. I just don’t know why it is! It seems like there needs to be some negatives somewhere to cancel things out. I could use some help figuring it out.

Thanks for all your help! I appreciate it!