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This 7 page Class Notes was uploaded by Dr. Elody Harris on Saturday October 3, 2015. The Class Notes belongs to MAT105 at California State Polytechnic University taught by TheresaVail in Fall. Since its upload, it has received 9 views. For similar materials see /class/218204/mat105-california-state-polytechnic-university in Mathematics (M) at California State Polytechnic University.
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Date Created: 10/03/15
Math 105 Section 42 Polynomial Functions De nition A polynomial function is of the form a fx 1 a 4xquot I ng39 le a I I n 7 llll where are real numbers and n is a nonnegative integer The domain of a polynomial function is the set of all real numbers NOTES The degree of a polynomial is the largest power of X that appears The zero polynomial function fX 0 ie all the coef cients in the above polynomial function are equal to zero is NOT assigned a degree Study Tip 1 Be able to identify polynomial functions See problem llO 2 Know which functions from functions to know are polynomial functions 3 Know that polynomial functions are continuous and smooth De nition A power function of degree n is a monomial of the form 3939 M where a is a real number a i 0 and n is a positive integer the Graphs class notes or Study Tip Be able to graph power functions using transformation techniques section 33 See problems 1926 De nition If f is a function and if a is a real number such that fa 0 then a is called a real zero of f synonyms root solution critical number The following statements are equivalent l a is a real zero of a polynomial function f 2 a 0 is an Xintercept of the graph of f 3 X 7 a is a factor of the polynomial function f TO FIND THE REAL ZEROS OF A POLYNOMIAL FUNCTION SOLVE THE EQUATION fX 0 De nition OR If a factor X 7 a occurs more than once in the factorization of a polynomial function then r is called a repeated root Once the polynomial is completely factored the power of the factor X 7 r determines the multiplicity of r If a factor X 7 a occurs In times in the factorization of a function fX then r is a zero of multiplicity m For eXample If fur quot7 2 39 51quot 4 The real zeros off are 2 7 5 4 and is a root of multiplicity 3 7 5 is a root of multiplicity 2 Note 4 is a root of multiplicity 1 NOTE Ifr is a zero of even multiplicity then the graph of the function touches the X7aXis at r If r is a zero of odd multiplicity then the graph of the function crosses the X7aXis at r De nition Theorem The points at which a graph changes direction are called turning points If f is a polynomial function of degree n then the graph of f has AT MOST n 7 l turning points If the graph of a polynomial function f has n turning points the degree of f is AT LEAST 11 1 For eXample fx TY X x 2 The degree of the function is 4 multiply out if needed It will have at most 3 turning points The real zeros are 0 l and 2 0 is a root of multiplicity 2 and the graph will touch the XaXis at 0 The graph will cross the XaXis at l and 2 NOTE Graphs of degree 4 will usually begin and end above the XaXis but this function has a negative sign out front and is a re ection about the XaXis so graph will begin and end below the XaXis Graphed in class on Oct 27 Theorem th END BEHAVIOR For large values of X the graph of the polynomial function r l 13x 1x IU 71 l 4 39 aux a resembles the graph of the power function fly X Study Tip Become familiar with Blue Summary on page 354 to be able to graph 3eciml Salve 17 com ple nj 39Huz Sim 4M4 zt ato 26 at 05 Wit lent 939P 5 WA 9444 musf be 1 5a divide boi sides byz 1 4 it Z Z 7 coeF ml t aw L tz39biz ciLz39 L 5 all swam L in 02g hbomsides quot a 1 Ll39m 1 2 m 1 23 t 39 hJ mV 4 use 57mm rwf Proferj i 25 N23 JG T q m 1 W23 1 E b0 t qt i fquotquotf L rL l Salva LY comF j u 53744211 550 5c c1 z 3 5c1toc 8 3 muh Fl ms BC 0L 2 5 Sudg rraL r 2 mm lacH Skies 02 Lg 35 bum x mjla Ex 3 Engage CoeFFiLEeut nP Lquot muS fL 1 z z to A w 5 L10 quot Mc 394 39 L 3 7 5 a z 3 LocFFicimd39 4 c fuw 239 c 39L 39 L i 50 add 1 meaHa CA Ey 35413973 sfd as 39km an a 62 55 we L W g ma slmPlf td use 5 j u aquot mat P referU LOGARITHMS MAT 105 SECUON 5253 Young Textbook Definition The logarithm of x with base b is defined as follows 39ll lOg X if and only if bl where b x are positive real numbers and b 1 1 Rewrite the following logarithms in exponential form logb36 2 IngmlOJWU 4 0992 l0g384 Evaluate the following logarithms Ll log4 logI 32 I6 logU 3 log3 I log 5 log128 logm 001 log O ProperTies of LogariThms p age 451 CompleTe The following LeT a and x be posiTive real numbers such ThaT quot l Then log 1 logquot I Memorize ProperTies of LogariThms Gold box on page 451 The ProducT Rule The QuoTienT Rule The Power Rule NOTE There is no properTy To expand The logariThm of a sum Know The Change of Base Formula on page 456 This formula is helpful To evaluaTe logariThms of any base wiTh a calculaTor MosT calculaTors have a common logariThm key or a naTural logariThm key If a 2 1 and M are posiTive real numbers Then 1 4M a 1019 0 lou M ln li logu and quot Ina EvaulaTe lOg 3 using boTh meThod s
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