Section. Monomials Objectives:. Multiply and divide monomials.. Simplify epressions involving powers of monomials.. Use epressions in scientific notation. I. Negative Eponents and Eponents of Zero A. Rules. For any non zero number a,. For any nonzero number a, 0 a = n a = B. Reminder. Negative eponents have nothing to do changes (e.g., making a number or.). Negative eponent a number (e.g., causes a number on the bottom to be moved to the and a number on top to move to the.) C. Simplify: y z = II. Multiply/Dividing Monomials A. Rule m n. Multiplying Powers with the SAME Bases: a a = m a. Dividing Powers with the SAME Bases: n a = B. :. Simplify the following epressions using the multiplication rule. 8 y. Simplify the following epressions using the division rule. a b yz a. b. a b y z. Simplify the following (Final answers need to have positive eponents) 8 - y ( y ) a. b. y y y p. Find the value of r that makes each sentence true. 8 r 7 r+ a. y = y y b. = m c. d. ( m ) m r r r+ = = r Precalculus Chapter Page
III. Scientific Notation A. Products and Quotients with Scientific Notation (Mental Math). Set : 0 ( ) (. 0 ) (.8 0 ). Set : a) In 99, billion telephone calls were placed by 0 million United States telephone subscribers. Write each in scientific notation and find the average number of calls placed per subscriber. b) The closest star to Earth (other than the Sun) is Alpha Centauri,. lightyears from the Earth. If one light year is.88 0 miles, how many miles from the Earth is Alpha Centauri?. Set : Simplify the following Mentally 9 (0. 0 )(. 0 ) (0.7 0 )(. 0 ) (. 0 ) (. 0 ) (0. 0 ) 9 (0.7 0 ) ( 0 ) 0 Homework: p., (8-)/, -8 all,, Precalculus Chapter Page
Goals:. To add polynomials. To subtract polynomials. To multiply polynomials Section. Polynomial I. Terms A. Polynomial B. Terms C. Degree of the Polynomial D. Eample Find the degree of the polynomials. +. 7y 7 y + 8 y y II. Adding and Subtracting Polynomials. Simplify: 8a ab + b + ab a b. Add: + + + ( 0 ) (8 ). Subtract: y + y + ( ) ( ) III. Multiplying Polynomial. 9 a (a 7 b ). ( + 8)( + ). ( + ). ( )( ). y y + ( )( ). ( + y)( + y 9 y) Homework: p. - all, - odds, -9 all Precalculus Chapter Page
Objectives:. Factor polynomials. Section. Factoring Polynomials I. Factoring Trinomials Factor the following trinomials (i.e., break them apart to be two binomials again). + 0 + 9 = ( )( ) =. 0 ( )( ) + =. 8 ( )( ) + =. ( )( ) =. ( )( ) II. Reverse of distributing (i.e., what is in common with all the terms?) Try the following. y 0 y = ( ) + y + =. ( ) A. What happens when both are together (reverse distribution and reverse FOILing)? Which do you do first? Always do reverse distribution first, then the reverse FOILing. Try the following. r + r r = ( )= ( )( ). r + r + 8 r = ( )= ( )( ). r r ( )= ( )( = ) III. Grouping Method Four or more terms Factor the following using the grouping method.. + 0 + a + a = ( )+( )= ( )+ ( )=( )( ). y 0 y + = ( )+( )= ( )+ ( )=( )( ). 0 0 a + = ( )+( )= ( )+ ( )=( )( ). 0 ( )+( )= ( )+ ( + = )=( )( ) Precalculus Chapter Page
IV. Difference/Sum of Two Cubes A. Formulas:. a b =. a + b = B. :. 8. 7 + What happens if none of the methods I use is able to factor the problem? Then the polynomial is prime. Try the following.. +. + 9. y Homework: p., (-0)/, 9, 0,, -0 all, 7, 8 Precalculus Chapter Page
Section. Roots of Real Numbers Goals:. To simplify roots having various indices.. To use a calculator to estimate roots of numbers. IV. General Information A. Definition of Square Roots For any real numbers a and b, if of b. B. Definition of n th n Roots For any real numbers a and b, if a C. Symbolism n. n is the. is the D. Note: Some numbers have more than one n th root.. a) What are the square roots of 9? b) What are the fourth roots of?. The nonnegative root is called the principal root. a) is asking for the root. a = b, then a is a square root = b, then a is a n th root of b. b) is asking for the of the root. c) ± is asking for the roots.. Odd indices a) 8 = This is the cube root. b) 8 = This is the cube root. E. Note: Roots be done over (E. ) V. Problems A. Find each root. a b. 8.. ( ) ± 9 8 + a. m n. 7. 0 y 8. ( ) 9. ( y ) 8 B. Calculator Approimations y. 97. 870. 87. The volume of a sphere is V = π r. Find the radius, if the volume is 8 ft. Homework: p. 8 (8-)/,,,, -70 all, 7 Precalculus Chapter Page
Section.A Radical Epression Goals:. To simplify radical epressions using multiplication and division.. To rationalize the denominator of a fraction containing a radical I. Radical Epressions A. Properties n n n. a b = ab. n n a b = n a b B. on Simplifying... 0a. ± y z 7 y z. a b 7 8 a b. 7. 8. 0.. m n. 0 8 mn 0. 0y 0 y 0y 8 y. II. Rationalizing the Denominator A. Protocol In math the proper is not to have any root in the. B..... 7 y 7.. Homework: worksheet Precalculus Chapter Page 7
Simplify the following using eact values only. Precalculus Worksheet Section.a Name:.. 00. 9. 0. ( 0 ). ( ) + 9.. 7 7. 9.. ( )( 7 8 ).. 9. 0 y n.. 7. 79 8. 70 9. y 7y 7. 7 8. 8 9a. 7 9 7. 0. 9t v tv 8 9. 0 8. 8 a b a b. 0 0 0. 9 9. y y Precalculus Chapter Page 8
Section.B Radical Epressions Goals:. To add, subtract, multiply, and divide radical epressions I. Adding and Subtracting Radicals A. Note:. To radicals they must be radical epressions (i.e., the and are ).. Treat like they are. B.. + + 7. 0 + 7 +. 7 7. + 0 a + a.. 0 7 II. Multiplying Radicals. ( + )( + ). ( + )( + ). ( + )( ). ( + 7 )( 7 ) III. Rationalizing the Denominator. +. + Homework: worksheet Precalculus Chapter Page 9
Precalculus Worksheet Section.b Name: Simplify the following using eact values only.. + 8 0. **. ( ) + 80. + 8. ** + +. 9 +. ** +. + + 8. ** 7. 7v t 8v t 7. ( ) 7 8. ( 7 + )( 7 ) 9. 8 +. ** 7. 7 + 8. + 8 0. 8 +. ** 9. 0. 8 + 0 0 + Precalculus Chapter Page 0
Section.7 Rational Eponents Goals:. To write epressions with rational eponents in simplest radical form and vice versa.. To evaluate epressions in either eponential or radical form I. Radicals in Eponential Form A. Rules:.. n b = b n m n m n b = b B.... 8. 9. 8. 7. 8. 7 7 II. Calculator. 98.. 0. 79 III. Simplify. a b.. w. ( ) y z. 0 8. a 7. a + b Homework: p. 0, -8 all, 9-7 odds, - all, 7-9 all, 7-80 all Precalculus Chapter Page
Section.8 Solving Radical Equations and Inequalities Goals:. To solve equations and inequalities containing radicals.. =. a + = a + 7. = 0. t + 7 =. 0 =. + = 0 7. + + = 8. = + 9. 0. +. = +. = 9 9 +. ( ) = 0. ( ) + + = Homework: p. 7, - odds, 9- all, - all Precalculus Chapter Page
Section.9A Comple Numbers Goals:. To simplify radicals containing negative radicands.. To multiply pure imaginary numbers.. To solve quadratics equations that has pure imaginary solutions.. To add, subtract, and multiply comple numbers. I. General Information A. Rene Descartes (00 yrs ago) came up with a way to solve Proposals:. i = where i is not a real number.. i = B. Pure Imaginary Numbers:. 8 =. =. Simplify: 8i i = and 0 = =.. Simplify: i = and i =. Solve: + 8 = 0 and a + = 7 0. Find the inductive resistance of a circuit if the reactance of a circuit is 7j ohms and the capacitive reactance is 7j ohms. (Note: Reactance of a circuit = inductive reactance capacitive reactance.) II. Comple Numbers A. Comple Number Form Real part Imaginary part. (8 + 7 i) + ( + i). (9 i) ( i). (8 + i)( i). ( + i)( i). Would two circuits in series with impedances of (8 j) ohms and (9 + j) ohms meet the code in of (0 j) ohms.. Homework: p. 7, 9-7 odds,, 9- odds,, 7,, 7, 8 Precalculus Chapter Page
Section.9B Comple Numbers Goals:. To simplify rational epressions containing comple numbers in the denominator. I. Comple Conjugates : + 7i 7i. ( )( ). ( 9 7i)( 9 + 7i) II. Comple Numbers in the Denominator Simplify i. i. + i + 7i. + 8i i. + 7i i. Refer to the application on page 7. Find the current if the voltage is ( 0 + 08 j) volts and impedance is ( j) ohms. Homework: p. 7 8- all,,, 7-7 all, 8-8 all Precalculus Chapter Page