Question

2. Use Rational Zero Theorem to determine the possible rational zero. Use Descartes’ Rule of Signs to determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros. Then find all zeros (real and complex of the polynomial.

Answer 1

l. Determine all factors of the Constant team and is a factor of the Constant team and a ipa theorem the rational use to find zero S019- rational heros :- The Rational zero teorem states that, if the polynomial + taxta Euesly crationeel q f(x) = anxent an-ian- has integer coefficients then 2010 of f(x) has the form of where, pisa factor of the Constant term as and qis a factor of the leading coefficient an! cohen the leading coefficileet 1, Heee Possible rational hero's are the factors of the Constauit team. Rational Zero How to find Rational Relaos by Hieorem : where P ) all factors of the leading coefficient 2. Determine possible, values of 2 q all factor of the leading coefficient Be Susie to include bottle Positive and negative Candidates. 3. Determine which possible 2090's wole actrice zero's by Evaluciting Sachcape of fP2)

Ex- use the rational SOI?:- f(u) are the Leading 2. possible rational Zerols of last team 1 and the factors of the the possible rational Zerols for! possible zold's are actual zoos by zelo fleeorem to find the grational Belos of f(n)= 27+72.44+1 The on quotients of the factors of the Co P-, by Substivity q + the constant team is the factors of love Beading coefficieet is a reel factors of 2 aske 9 = £1,? if aney of the possible values for & are & 1 and the function we can determine. cohich of. Hee Hliegé Values of * en fra) f(-1) = 12 (-1134 (182 - 4614 - 2 +144+1 4 f(!) = 20)3+ 09-.40.2+1 $00). 2+1+1 -4 = 0 =0 2) 11%) 3+ (%) 41-\) + f(-1) 3 2 f (1/2) 6)+()2- 41%29+1= -Va out of those w,-/28% are not geros of fo. 1 is the only reational aero of f(x).

to Descartes Srceles of signs, Descartes Rules of signs +91.tao be a if we let from = Anx?taniany Polynomial function wille steel coefficiently • Mie nomb egri oli positive gecel zero's is sittiese Speal to the nombee of sign changes of fra) (ov) legs teen the number of sign changes by an Even integee. ② the number of negative deal zeros is either equal to the number of Sign changes of fr-xor) is less than the number of sign Changes by an Even integer

negative Zeros this tells us that the function must have 78. a semilati relationship between thee use Descrates! Rules 3 of signs: Theone is a straight forward way to determine the possible number of positive and for any polynomical function. If the polynom al is coliten in de scending order, "bescentes 'Rules of signg teug U8.0f a relationship between teie number of Sign changes in fox) and the number of Positive seal Zeno's for example , the polynomial function below has one sign change ） I positive recel zero. number of Sign changes im of 6-20) and see hombere of negative real zergs In cafe , f (-a) hap 3 sign changes their tells us teat fears could have 3 ori negative real perge

So there wee Edit f(x)= -x - 3x3 + 6x² ux-12 flee number of sol? Begin by determining Sign changes. all = f(n) - 32 1 3+ 6x²_ux-12 СЯ СЯ Either 2 (or) o positive gecel roots Next we examine f(-x) to determine the number geal roote of negative

A F(-X)= -(~204.–3(-2)3+66x12–460X) -12 f(x) = -24 +383 +6+48-12 f(-x) = -x4+373 +68++48-12 Again, there are two sign changes, so there are titleen 2(07) o negative recel roots Millee are foile poffibilities, as we cansee below table : Reditus Complex total zelos Zero zes Negative real gueell zeros O 4 2 2 2 ee 4 4 2 o 4 4