PRECALC ALG & TRIG
PRECALC ALG & TRIG MAC 1147
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This 33 page Class Notes was uploaded by Marquise Graham on Saturday September 19, 2015. The Class Notes belongs to MAC 1147 at University of Florida taught by Larissa Williamson in Fall. Since its upload, it has received 12 views. For similar materials see /class/207048/mac-1147-university-of-florida in Calculus and Pre Calculus at University of Florida.
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Date Created: 09/19/15
L3 Rational Expressions nth Roots Rational Exponents Rational Expressions The domain of an expression in one variable is the set of all real numbers for which the expression is defined A rational expression is a ratio of two polynomials The domain of a rational expression is the set of all real numbers which do not make the denominator equal to zero Reducing Rational Expressions z if 97200720 bc 9 Example Find the domain and reduce the expression to lowest terms x25x6 1 3x9 Domain 25 x2 25 5 x 2 Domain Multiplication and division biQdio i bchiQdio 0 Example Perform the indicated operations and simplify Give restrictions on the variables f 8 4y 2 y3 y2 5 y 6 26 xZ y2ltx ygt2 2 90 90 Addition and subtraction In order to add subtract rational expressions we use the Least Common Multiple LCM of the denominators To Find the LCM of the Denominators 1 Factor polynomials that are in the denominators 2 The LCM is a product of all different factors of the denominators numbers variables expressions each raised to the largest power that appears on that factor 27 Example Add or subtract as indicated Give all restrictions on the variables x x 3 3x2 x 2x3 x2 Note Be aware of the case when the denominators are additive inverses of each other Example Perform the indicated operations 28 Mixed Quotients A mixed quotient complex fraction is a quotient of rational expressions Simplifying a Complex Fraction Method 1 Multiply both numerator and denominator by the LCM of the denominators of all simple fractions and simplify Method 2 Perform the indicated additions andor subtractions in the numerator and denominator and then divide Example Simplify the complex fraction mixed quotient x xl 2X1 X 29 nth Roots A number I is an nth root of a number a if bquot a The principal nth root ofa real number a is denoted lt5 n 2 2 and defined as follows If n is an even number and a 2 0 then lt5 is the nonnegative nth root at lt 0 then lt5 is undefined not a real number If n is an odd number then lt5 is the only real nth root of a number a and it has the same sign as a 30 Note Since Qg is an nth root then 5 01 Example Evaluate the radical expressions 327 16 Cancellation Rule for Exponents and Radicals Wzla ifn22 andniseven Qa nza ifn23andnis0dd m Wlx not ix 31 m Simplify the expressions 4 34 6 26 Rules for Radicals we assume that all radicals are defined mpg Simplify each expression VF s sE 32 Simplifying Radicals 1 Use the cancellation rule for radicals and exponents to remove all possible factors out of the radical When it is done the factors left in the radicand will have smaller exponents than the indeX of the radical 26 2 2 Rationalize the denominator 3 Reduce the indeX of the radical as far as possible 4x2 ifoO 33 Example Simplify the expressions with radicals Assume that all variables are positive when they appear 324x5y7 4 32a5b7612 34 Caution x y 72 J J le y2 2 x y Addin and Subtractin Radical EX ressions We can only add or subtract like radicals Like radicals have the same index and the same radicand Example Add or subtract as indicated JEJ8 3 48x2y x l75y 33y xgt0ygt0 35 Rational Exponents If a is real and n 2 2 is an integer then a 4 provided that 3 is defined Note If n is even and a lt 0 then 4 and ai are not defined Example Simplify the expressions if possible 1 1445 1 1442 8g 36 If a i O is a real number and m and n are integers containing no common factors with n 2 2 then provided that lt5 is de ned Example Evaluate each expression 125g 1 E 16 37 Example Simplify the expressions Assume that all variables are positive when they appear 48x8 2 y 5 V3 5902 J5 NE z 38 Example Write each expression as a single quotient in which only positive exponents or radicals appear 2 Vx2l x x2l x2l 2 l l l x 3cl2 c3cl 2 x 0x l 39 L17 Rational Inequalities Zeros of a Polynomial Function Solving Rational Inequalities 1 Get all terms on the left side of the inequality with a O on the right side and simplify the lefthand side into a single fraction Write the domain Reduce the fraction 2 Find all real zeros of the numerator and denominator Determine their multiplicities 3 Divide a real line into intervals using the w found in Step 2 and the numbers that are not in the domain Label an endpoint as o if it is to be included in the answer and label it as o if it is not Note Zeros of the denominator are never included Zeros of the numerator which are in the domain are included if and only if the inequality is non strict S Z 4 Use the end behavior of the polynomials in the numerator and denominator to find the sign of the fraction on the rightmost interval when x gt 00 5 Set the signs on each other interval by moving from the right to the left and changingnot changing the Sign depending on multiplicity 6 Select the intervals with the desired sign of the function according to the inequality in Step 1 182 Important Never multiply 0r divide both sides of an inequality by an expression that contains a variable and varies its sign depending on the variable x 22 x 13 Example Solve gt 0 183 x 1x22x3 lt O E pl 81 mm 6 OVCX 33x2x3 184 Example Solve the rational inequalities 5 3 lt k2 k 4 5 3 185 Horizontal Asymptotes and Limiting Size of Population 50 Manatees are taken to a river sanctuary The population of the manatees is given by NI105 41 l 0051 Where lis time in years 120 Find the population after I 10 years after 100 years What is the limiting size of the population as time increases innn Zeros of a Polynomial Function Division Algorithm for Polynomials If f and g are two polynomials and g is not the zero polynomial then there are unique polynomials q quotient and r remainder such that f qltxgt r or fx mm roe gx gx where rx either the zero polynomial or of degree less than the degree of gx Note If gx x c then fx qxx c r where r is a number Ifxcthen fc Remainder Theo rem If a polynomial f x is divided by x c then the remainder r f c 187 Example Find the remainder if f x x4 6x3 2 is divided by x 2 Use synthetic division the Remainder Theorem Example Use the Remainder Theorem to find f 3 if fcc6 6c5 54x2 16x1 188 Note If the polynomial f x is divided by x c and the remainder r 0 then fc O that is C is a zero of f x Example If fx x3 6x 4 is 2 a zero of f Factor Theorem The polynomial x c is a factor of the polynomial f x if and only if f c O Proof If x c is a factor of f then f x 2 hence f c Iffc0thenrfc andfx therefore x c is a factor of f 189 Example Is x 1 a factor of x3 2x 1 Example Factor f x into linear factors given that C is a zero of f x fx3x3 5x2 16x12 c 2 190 Fundamental Theorem of Algebra Every polynomial of degree 1 or more has at least one complex zero Let degfxn n21 Zeros F1 f X X mice r2 f X 5 fltxgt Thus fx anx r1x r2x F Number of Zeros Theorem A polynomial of degree n has at most n distinct zeros Note A polynomial of degree n has exactly n complex zeros if to count each zero as many times as its multiplicity Conjugate Zeros Theorem If f x is a polynomial whose coefficients are real and if a bi is a zero of fx with a and 9 real numbers then a bi is also a zero of f x l9l Example One zero is given find all others x3 6x211x 6 3 x4 10x3 27x2 10x 26 192 Example Find the polynomial of degree 3 with real coefficients that satisfies the conditions zeros 2 1 O f 1 1 193 Example Find a polynomial of the lowest degree possible with only real coefficients which has the given zeros l 6 3i 194 Intermediate Value Theorem If f x is a polynomial with only real coef cients and if for real numbers a and b the values f a and f b are of opposite signs then there exists at least one real zero between a and b Example Show that the polynomial has a real zero between 2 and 3 fx 2x3 9x2 x 20 L26 The Inverse Trigonometric Functions The six trigonometric functions are not onetoone on their domains To de ne the inverse of a trigonometric function we consider the function on the restricted domain Where it is onetoone The restricted domain we will call the Interval of De nition Intervals of De nition and Inver e ofthe T 39 Functions fx sinx Interval afDe nitian 1 x sin l x Domain Range zx ote f 1 x sin l x is an add function 312 f x cosx Interval of Definition 1 39 quot lt i l5 X 1 71 1 39 f x cos x Domaln Range f x tanx Interval of Definition f 1 x tan 1 x Domain Range Note f 1x tan lx is an odd function f x cotx Interval of De nition f 1x cot 1 x Domain Range Notes The domains of the inverse trigonometric functions are the ranges of the corresponding trigonometric function The ranges of the inverse trigonometric functions are the Intervals of Definitions of the corresponding trigonometric functions Important Values of an inverse trigonometric function are the angles from the Interval of Definition see the ranges of the inverse functions
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