Why does the quadratic factor to (x -r) (x-s) automatically?

[RobertO770]

Hi!

On DS 99 p. 283 the question is:

[deleted because problem is from a banned source - see below]

The explanation automatically puts forward that the factors are both negative. But could they also be both positive? In that instance rs & c < 0 are not impacted as answer solutions.

However, evaluating statement one becomes harder (which is where I went wrong)

Hope you can clarify the reasoning for me. = )

[tim]

OG is a banned source; it is illegal to post OG questions anywhere on the web. If you are in one of our classes, please ask OG questions during office hours or before/after class.

Please take a closer look at the explanation though; it does not say that the roots are negative. If r and s are roots of a polynomial, then (x-r) and (x-s) are factors. This is a universal definition, and in no way indicates whether r and s are positive or negative.

[RonPurewal]

On a related note, one of the most common points of confusion here is the fact that things like "-x" can represent positive quantities. I.e., if x itself is negative, then -x is positive.

Lots of people try to do things like, "Well, if -x is positive, then I'll just take off the '-' sign!" Nope, can't do that. You have to understand that "-x" just means "the opposite of x", whether that be positive, negative, or zero.
Same thing with stuff like "x - r" or "x - s". If r or s is negative, then these things will be greater than x ... but that's not a problem; it's just the way algebraic notation works.