2.4Polynomial and Rational Functions Polynomial Functions

1 Polynomial and Rational FunctionsPolynomial FunctionsGiven a linear functionf(x) =mx+b, we can add a square term, andget a quadratic functiong(x) =ax2+f(x) =ax2+mx+b. We cancontinue adding terms of higher degrees, we can add a cube termand geth(x) =cx3+g(x) =cx3+ax2+mx+b, and so (x),g(x),andh(x) are all special cases of a Polynomial ( Polynomial Function)Apolynomial functionis a function that can be written in theformf(x) =anxn+an 1xn 1+..+a1x+a0forna nonnegative integer, called thedegreeof the coefficientsan, an 1, .. , a1, a0are real numbers withan6= that althoughan6= 0, the remaining coefficientsan 1, an 2.

2 , a1, a0can very well be of Polynomial FunctionThe domain of a Polynomial function isR, the set ofall domain off(x) =xnisRregardless the value ofn(any nonneg-ative integer), and so is the domain ofg(x) =axn, whereais somereal number. Clearly, if you add, sayk, such Functions with differentdegrees (n) the domain of the resulting function will still 2. Functions and Polynomial and Rational FunctionsConsider a functionf(x) = (x 1)(x 2)(x 3). It could be rewrittenasf(x) = (x 1)(x 2)(x 3)== (x 1)(x2 2x 3x+ 6) == (x 1)(x2 5x+ 6)==x3 5x2+ 6x x2+ 5x 6==x3 6x2+ 11x ,f(x) is a Polynomial function of degree :How manyyintercepts doesf(x) have?

3 Answer:Only one,y=f(0) = 6. Any function can have at mostoneyintercept, otherwise it will not pass the vertical line of a Polynomial FunctionIff(x) =anxn+an 1xn 1+..+a1x+a0is a Polynomial function,it hasexactly oneyintercepty= :How manyxintercepts doesf(x) have?Answer:f(x) has 3 intercepts. 0 = (x 1)(x 2)(x 3) = x= 1orx= 2 orx= of a Polynomial FunctionA Polynomial of degreencan have,at most,nlinear , the graph of a Polynomial function of positive degreencan intersect thexaxis at mostntimes. Thexintercepts off(x) =anxn+an 1xn 1+.

4 +a1x+a0could be found by solvinganxn+an 1xn 1+..+a1x+a0= 2. Functions and Polynomial and Rational Functionsxy 7 6 5 4 3 2 101234567 7 6 5 4 3 2 101234567f(x)Consider a functionh(x) = (x2+ 1)(x 2)(x 3).h(x) = (x2+ 1)(x 2)(x 3)== (x2+ 1)(x2 2x 3x+ 6) == (x2+ 1)(x2 5x+ 6)==x4 5x3+ 6x2+x2 5x+ 6==x4 5x3+ 7x2 5x+ (x) is a Polynomial function of degree 4, but has just 2xintercepts,because the equation 0 = (x2+ 1)(x 2)(x 3) has just 2 roots (zeros),which arex= 2 andx= 2. Functions and Polynomial and Rational Functions 101234 448121620xyh(x)Note thatf(x) =x3 6x2+ 11x 6 has degree 3, which is anoddnumber.

5 It starts negative, ends positive, and crosses thexaxisoddnumber of (x) =x4 5x3+ 7x2 5x+ 6 has degree 4, which is starts positive, ends positive, and cross thexaxisevennumber (x) = f(x) = (x3 6x2+ 11x 6) = x3+ 6x2 11x+ 6,andn(x) = g(x) = (x4 5x3+7x2 5x+6) = x4+5x3 7x2+5x 6. 7 6 5 4 3 2 101234567 7 5 3 11357xym(x) 101234 20 16 12 8 448xyn(x)4Ch 2. Functions and Polynomial and Rational FunctionsDefinition (Leading Coefficient)Given a Polynomial functionf(x) =anxn+an 1xn 1+..+a1x+a0,the coefficientanof the highest-degree term is called theleadingcoefficientof a Polynomial functionf(x).

6 Graph of a Polynomial FunctionGiven a Polynomial functionf(x) =anxn+an 1xn 1+..+a1x+a0:(a) ifan>0 andnis odd, then the graph off(x)starts neg-ative,ends positive, and crosses thexaxisoddnumber oftimes butat least once;(b) ifan<0 andnis odd, then the graph off(x)starts pos-itive,ends negative, and crosses thexaxisoddnumber oftimes butat least once;(c) ifan>0 andnis even, then the graph off(x)startspositive,ends positive, and crosses thexaxisevennumberof timesor does not cross it at all;(d) ifan<0 andnis even, then the graph off(x)startsnegative,ends negative, and crosses thexaxisevennumber of timesor does not cross it at :(c) is a reflection in thexaxis of (a), and (d) is areflection in thexaxis of (b).

7 Also note that a Polynomial function always eitherincreases or de-creases without boundasxgoes to either negative or positive 2. Functions and Polynomial and Rational FunctionsContinuity and Smoothness of Polynomial Func-tionConsiderf(x) =2|x| (x) has a discontinuous break atx= 0. 5 3 1135 4 224xyf(x)6Ch 2. Functions and Polynomial and Rational FunctionsConsiderg(x) =|x| (x) is continuous, but not smooth due to asharp corner at (0, 2). 5 3 1135 4 224xyg(x)Considerh(x) =2x (x) has a discontinuous break atx= 1. 5 3 1135 5 3 1135xyh(x)Graph of a Polynomial FunctionThe graph of a Polynomial function iscontinuous, with no holesor breaks.

8 That is, the graph can be drawn without removing apen from the paper. Also, the graph of a Polynomial is smooth , has no sharp 2. Functions and Polynomial and Rational FunctionsRational FunctionsJust as Rational numbers are defined in terms of quotients of integers, Rational Functions are defined in terms of quotients of ( Rational Function)Arational functionis any function that can be written in theformf(x) =n(x)d(x), d(x)6= 0wheren(x) andn(x) are example,f(x) =1x, g(x) =x 2x2 x 6, h(x) =x13 8x5,p(x) =x4 5x3+ 7x2, q(x) = 123, r(x) = 0are all Rational (x) andd(x)

9 Are polynomials , then they both have ,Domain of a Rational FunctionIff(x) =n(x)d(x)is a Rational function, then its domain is the set ofall real numbers such thatd(x)6= 2. Functions and Polynomial and Rational FunctionsExample 1 Find the domain off(x) =x2+1x2 7x+109Ch 2. Functions and Polynomial and Rational FunctionsVertical and Horizontal AsymptotesRecall that a Polynomial function is alwayscontinuousand smooth .It is also true that ifxincreases or or decreases without bound, thenfunction also increases or decreaseswithout bound.

10 However, thismay not be true for a Rational function. Also, a Rational functionmay not have a Rational functionf(x) =x 3x 2. Its domain ( ,2] [2, ),or all real numbers except forx= 2,xf(x) 2= 2= 2= 2= 2= 2= 2= 2= 2= 2= 2= 2= 2= 2= 2= 2= 110Ch 2. Functions and Polynomial and Rational Functionsxf(x)-100000 100000 3 100000 2= 100003 100002= 10000 3 10000 2= 10003 10002= 1000 3 1000 2= 1003 1002= 1000 3 1000 2= 1003 1002= 10 3 10 2= 13 12= 5 3 5 2= 8 7= 30 2= 3 2= 2= 2= 133 33 2=01= 055 35 2=23= 310 2=78= 3100 2=9798= 31000 2=997998= 310000 2=99979998= 3100000 2=9999799998= 8 6 4 202468 8 6 4 22468xyx= 2y= 1 The graph off(x) gets closer tothe linex= 2 asxgets closer to 2 is avertical asymp-toteforf(x).