Finding zeroes and determining multiplicities help please? (Picture below)

Question

anonymous · Answer

attachment

jhannybean · Answer

factor out an $x^2$

kittiwitti1 · Answer

I'd say use synthetic division... c:

mathmate · Answer

Yes, start by factoring out $x^2$, by which time we see that it becomes a quadratic in $x^2$, so substitute y=$x^2$.  Solve for y by factoring or quadratic formula, and eventually solve for x=$\pm\sqrt y$.  If the root of y is negative (say -a), just write the factor $(x^2+a)$ as a factor.
After factoring the whole expression, see which ones are simple roots.  They cross the x-axis.  The factor(s) like $(x^2+a)$ don't have real roots, so don't cross the x-axis.
Those with multiplicity greater than one such as $x^2,(x-b)^3$ will either cross or touch the x-axis. Roots with even multiplicity will touch the x-axis.  Roots with odd multiplicity greater than one will touch but cross the x-axis.  See drawing below.|dw:1419338985602:dw|