@amistre64 Can you please explain the Rational Roots Theorem? Example: List the possible rational roots of 3x^3+5x^2-2x+1=0

Question

amistre64 · Answer

yay!  no spinning wheel of death ...

anonymous · Answer

Lol?

amistre64 · Answer

do you recall a method named "FOIL"?

anonymous · Answer

YES :)

amistre64 · Answer

the key letters in that is the F and L ... at least for this purpose

amistre64 · Answer

assume we have some factors:$$(ax-b)(cx-d)$$or expanded to be:$$(ac)x^2+(-ad-bc)x+(bd)$$
have you gone over the quadratic formula yet?

anonymous · Answer

Yes I have.

amistre64 · Answer

if we run that thru the quadratic formula; we would get zeros that are formed from the factors of the last term, $bd$, divided by the factors of the first term $ac$.  This works for any polynomial in general.

amistre64 · Answer

so if there is a rational solution to this setup, it would have to be a product of:$$\frac{factors~of~last~term}{factors~of~first~term}$$

anonymous · Answer

Okay, so I run the equation through the quadratic formula and... My solution is supposed to look like (factors of last term)/(factors of first term)?

amistre64 · Answer

the quadratic only works for degree 2 polynomials ... but its a way to help visualise the rational roots thought process

amistre64 · Answer

3x^3+5x^2-2x+1=0

last term is 1;  factors of 1 are:  1
first term is 3;  factors of 3 are:  1,3

if there is a rational root to this setup, it will have to be from a pool of options:$$\pm\frac{1}{1},\pm\frac{1}{3}$$

anonymous · Answer

Oooooh. That's not as complicated as I thought...

amistre64 · Answer

yeah, it is a pretty simple idea :)

anonymous · Answer

So for x^3+2x+3=0... I do the same thing? First and last factors?

amistre64 · Answer

yep

anonymous · Answer

So I have the same answer then.

anonymous · Answer

Huh-uh... No. Sorry.

amistre64 · Answer

not quite.

last/first

factors of last are 1,3
factors of first are  1

so the possible solutions are:  +- 1,  +- 3

amistre64 · Answer

the rational roots creates a set of possible solutions, but it does not guarentee that any of them are actually a solution.

IF it has a rational solution, then it would have to be one of the options in the pool; but there is no certainty that there is a rational solution to begin with.