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By Harley Flanders; Justin J Price

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It is important how many times each factor (x - r1) occurs, so we write r, < '2 < . . < r, • to show clearly each zero r1 with its multiplicity m1 • 8. Polynomials and Rational Functions 37 x Ff&. 3 Ff&. 4 Graph of y = (x - r)2(x - s� r>s Graph of y = (x - r)(x - s)(x - t� r

Distance Formula The distance between J(x 2 - x a )2 + (Y2 - (x a , Ya ) and (x2 , y2 ) is }1a)2 . 1 . FU NCT I O NS A N D GRAPHS 44 ,- I I I ( \" 1 • r-1 --� '1 ) r-1 ___ 4 , ,� · ' • , (\ � I I l (\I ' I F'11- I I l I , • Distance between points horizontal line or the sameanverti cal line,point. the If (x is>'iI)xand- (xx 2I, ory2) Iliey1 onytheI ;same distance there is no need to introduce auxiliary 2 2 Nevertheless, the distance formula still yields thediscuss correcta fanswer. applicati o ns of the di s tance f o rmula, let us problems, that is, As problems of finding all points in the plane that satisfy certaiewnlocm geometri c conditions.

1· = 41 Y = y=x+3 +2 (x + 1)2 - 2 (x + l)(x - 2) (x + 2)(x - 3) (x + 2)x(x - 2) (x + l)(x - 1 ) .... y = 43 Y = = (x + l)(x - 2) (x + I )(x - I ) (x + 2)x(x - 2) 4+2 x2 + 2 = x(x -x1 )(2x2 + S) 46 J' = (x - l)(x + l)(x2 + 1) x2 + I - 2)3 48 = (x y = x(x + J)(x2 + 4) (x + 1)2 " Suppose r(x) = f (x)/g(x) is expressed in lowest terms and deg/(x) = I + deg g(x). Why does the graph of y = r(x) have an oblique asymptote? Under what circumstances does the graph of a rational function have a horizontal asymp­ tote?

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