daman's g{iyt 2*1*2=2 =5zg#=5r± 3 S shit exponent ? 25-3 a m * a n 5. I base =/ 5 r Properties of Exponents Terminology In a b a is the b is the

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1 I Properties of Exponents Terminology In a b a is the b is the " a to the bth power " base exponent The Product Rule for Like Bases daman's 3 * z z a m * a n *1*22? 253 am@ a m an Ex: Simplify 1) (5r 2 st 3 )(3rs 2 t) 53 mrss 't 't / 5 r r 3+1 g{iyt 3 2) 5r2 st 3 3rs 2 t shit ' 5zg#5r± 3 S Other Exponents Zero and Negative Integers

2 son,by gef was y z z z ' z ? " 22???? 23?? d 2? 2 2 F 4 # a 0 I except 00 is undefined Ex: Evaluate 1) 3 0 I 2) π 0 3) (5) 0 : I 4) ) 0 0 undefined

3 For any nonzero real number a and integer n a n 1 a n and a n 1 a n Ex: Express using positive exponents and then simplify, if possible 1) 5 2 2) 3 1 3) z±s 's 's ) (2) 2 it # 5) 2 2 6) 12x 3 y 6 7) ab 3 c 4 d 5 f ) or 9) Y I 525 5*45 E 5 or 5j The Power Rule oddest, Has

4 The Power Rule (3 2 ) 4 : IK5E ;i ss :3 :* (a m ) n a mn (as long as a m exists) Ex: Simplify 1) (2 1 ) 2 zt 's tk 224 2) (x 4 ) 5 C4) 5, Powers of Products (ab) 3 ( a b) Cab) ( a b) a3 b3 (ab) m a m b m ( a b )m am b m b 0 Ex: Simplify e #f, j 1) (2a 1 b 3 ) 2 (2a 1 b 3 ) 2

5 2) ( 4x4 y 2 6x 1 y 4)4 t hyyj4 Ee Y C 25 ' ( 5j he F 3) ( 8xy2 z 2 4xy 1 z 4) 2 *n :* IE?IjyiYIf(zy3z2)2z2(j3jyEj2z2y6z4zy# y z4 Scientific roadstead Notation

6 5 2 Scientific Notation First, some review: In ' of lot Scientific Notation: X Useful for very big or very small numbers: Ex: Convert to Scientific Notation 1) 35 2) µ 3 3) ,flK3'2"n98y65y}zTMhM 4) YYYYYYYY,y y / ] X / O 15 Ex: Convert to Standard Notation 1) 1036 X ,600 um big #

7 2) 1036 X ) 1036 X small # I muumuu 1 0,360,000,000,000 Ex: Simplify and write the answer in Scientific Notation 1) 42 X X C 3) ) ( )(3,500,000,000) xxxx (61 109), ( ) Move Move Left Right ( 6 1) ( 3 5) ( * in ' siii n or Notation 2135 Standard

8 Polynomials Polynomials are expressions built out of a combination of,, +,, constants and/or variables Examples of polynomials: 25 +6,5 * 4, Monomials Polynomials with one term, xy * xfxys ',5 y Monomial x 3x 2 x 3 14xy 4 6 Degree o} exegetes Binomials Polynomials with two terms Leading Binomial Degree Term x + 1 3x 2 + x 3x 14x Trinomials Polynomials with three terms 3 3 Leading Trinomial Degree Term x 3 + 3x x 2 y 4 16x differed Order Ex: 4x 4 3x 3 + 6x 2 1 F Degree 4 "

9 X are Area X xx2 590 " squared " x Volume \ 3 xxx " x cubed "

10 Descending Order (Traditional) ' I Ascending Order It What if there are two variables? Ex: 4x 2 y 4 16x It depends on the setting For now, you can arrange in descending powers of x: y 412 Polynomial Functions Ex: P(x) x 2 + 4x 1 Find P(5) x ' +4 1 (5) ' +4 E) I P(5) C5) 2 t 4( 5) Adding and Subtracting Polynomials Ex: Add or Subtract, and Combine Like Terms:

11 1) (3x 3 2x 4) + (5x 3 + x 2 10) it / am 2) (9r 5s t) (7r 5s + 3t) 9r 5s 2r+Os t 7rt5 4t Zr 4t

12 54 Multiplying Polynomials Multiplication of Monomials Ex: (8x 3 y 2 t)(2xy 3 t) C 8) Multiplication of a Monomial and a Binomial: Ex: Multiply out: 3a 2 b(a 2 b 2 ) fdx3xy2y3ttl6x4y5ta3a2boi3a2bb23a4b3a2b3tx@icxyx22xtefxzfdxeeetezxxzx Multiplication of Binomials WITHOUT the F Word Ex: 1) (x + 1) (x 2) ' * z 2) (y 3 5) (2y 2 + 4) 3 (2,12+4)5/2144) 2y5t4y3 y?2y' + PY Dy ' 20

13 LONGER Polynomials Ex: 1) (p + 2)(p 4 2p 3 + 3) ( p4gs#tt(p42p3t3)p52pyt3p+2py4p3t6 a 0 o 5 p terms 3 terms 2) (5x 3 + x 4)(2x 2 + 3x + 6) 9 terms ) ( ) ) ' ioxitisxitzsix3li#6x:zi

14 THE Shortcut FOIL Ex: Multiply (x + 2) (x + 3) First Outside Inside Last M } xx x 3 Zx 23 FIOL two c ' Ex: Multiply using FOIL: LFO F 0 1 L 2xfD 1) (2x + 4) (x 3) : 2 2 2) (3x 3) (x 2) 3) (x + 2) (x + 2) 2 (3) 4tD 4t3 ) ( +z) / ) (x + 3) (x 3) X ± 9 More Formulas: Square of a Binomial (A + B) (A + B) * Product of a Sum and a Difference IEEEtt#kA2AB+BAB2 (A + B) (A B) (A+B)CAB)tA2B [ Difference of Squares

15 B) Ex: Multiply ( A 1) (1 4x) (1 + 4x) 2) (1 4x) 2 B) ( At B) A ( 15 ( ( 1 4x)( ( A B) ZABT ( A A2 4x+( ) (2a 4 + ab) (2a 4 ab) Gay ' Cab ( )24a8a2b A + B) ( A B) A ' 132 Taibl ) 4) (2a 4 + ab) 2 Gay '+2GaD(ab)+Gb54a8+4a5b ( A +135 A2t2ABtB2 ' 5) (5y i 5x) (5y + 4 3x) [ (5 +4)+3 Ex: Given f(x) x 2 4x + 5 Find: f ( ) x ) f(a) + 3 x][ey+d D(5yt4)?G [ ( 5y)2t2(5y)(4) +42 ( A + B) ( A B) A ' 132 ( ) f(a + 3) 3) f(a + h) f(a) a24at5t3a2_4at8fcdx24xt5cats54catdt5a2t6at94al2t5a2t2at2aeathi4etdtsjfa24atdfitza_hh2e4an4htda2t4a5a2t2ahth24a4ht5a2@h2hot4a 5

16 ( A B) ( A B) A2 AB A2 : AB ( A B) ' A ' 2AB+B ( A+B)2 A2+2AB+ BA+B AB+B ' ' '

17 2 z 55 Common Factors and Factoring by Grouping Preliminaries: an 3(x + 2) x ( x + 2) In this example, 3 is the common factor Ex: Factor out Common Factors 1) 2x 10 zz 5 2) 2x 10 2 ( 2 5) ( x + 5) 3) 2x + 10 chezttfsj 2 ( 5) 2 x I 4) 6t 2 12t Gt A ) or 6 ( E 5) 3y 7 y 6 y 2 a) 6 t Ct 2) y 43 y 5 wee y 4 I) 6) 2x 3 6x x x ( ) 7) 100m + 33 I ( 00 m + 33) Tiuftrivial Prime Polynomial

18 7 8 9,67222 ± } 'Ee,, z 04 5 ) * x 5) O Moykftf 2 ( 5) x ) Factor

19 8) 13b 2 c 3 26b 2 c l3b2c(c2 2) 9) 13b 2 c 3 26b 2 c + 2b b ( 13 be 26 bet 2) 10) 25t 10 u 2 v 5 10t 7 u 2 v 4 55t 6 u 2 v 4 5t6uiv4(5E' v Zt ID

20 E 1) 1) l Factoring by Grouping Ex: Factor completely 1) r (t 3) s (t 3) ( Ct 3) or ( r s ) Ct 3) 2) 4x x 2 3x 15 4 terms :Yi i s F YI ' n' Try ) t 6 1 t 5 + t 6 D t(t4 # I) Try : the t5(t Ect Itt + t I +14 (t5+dct D )

21 56 Trial and Error Nonbook method Ex: Factor Completely F 01 L 1) x 2 + 5x + 6 ( D)( D Yitxo * 62? s0 t' 2) y 2 9y + 20 ( y ) ( y ) 9y 20 C 4) f5) ( to ) C 2) ( y 3) x 3 x 2 30x 5y+zo loy +20 (45*4)424 io) ( y 2) y2 Zy C 20 ) C 1) ( y 20 ) ( y yky D ZOYTZO : 4) x 2 x 7

22 5) 6x 2 19x ) 6x 6 19x x 4

23 57 Difference of Two Squares, Sum and Difference of Two Cubes Remember that (A + B) (A B) A 2 AB + BA B 2 A 2 B 2 This gives us the formula: A 2 B 2 (A + B)(A B) Difference of Two Squares Ex: Factor Completely 1) x 2 9 2) 25 y 6 49 x 2 3) 5 5p 2 q 6 4) x Sum and Difference of Two Cubes (A + B)(A 2 AB + B 2 )? (A B)(A 2 + AB + B 2 )?

24 A 3 + B 3 (A + B)(A 2 AB + B 2 ) A 3 B 3 (A B)(A 2 + AB + B 2 ) Sum of Two Cubes Difference of Two Cubes Ex: Factor 1) x ) 125x 3 8 3) 128y 7 250x 6 y

25 58 Summary of Factoring Techniques Key Count the Terms 1) Greatest Common Factors Always try this FIRST 2) Two Terms Sum of Squares A 2 + B 2 Can t be factored Difference of Squares A 2 B 2 (A + B)(A B) Sum of Cubes A 3 + B 3 (A + B)(A 2 AB + B 2 ) Difference of Cubes A 3 B 3 (A B)(A 2 + AB + B 2 ) 3) Three Terms Trial and Error Perfect Square 4) Four Terms Grouping 5) Keep factoring until: 6) Check by:

26 Ex: Factor Completely 1) 10a 2 x 40b 2 x 2) x ) 7x y 2 4) 2x a 2 20ax

27 5) 12x 2 40x 32 6) 3x ax 2 + 4ax 7) 42ab + 27a 2 b 2 + 8

28 59 Solving Equations by Factoring Quadratic Equation: An equation in the form ax 2 + bx + c 0, where a, b, c represent real numbers and a 0 Zero Factor Property: ab 0 iff a 0 or b 0 Ex: Solve 1) x ) x 2 x 6

29 3) If f(x) 3x 2 4x, and g(x) x 2 + 2x 4 find all a such that f(x) g(x) 4) Find the domain of f if f(x) x 2 x 2 + 2x 15