MATH 90 CHAPTER 5 Name:.

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1 MATH 90 CHAPTER 5 Nme:. 5.1 Multiplictio of Expoets Need To Kow Recll expoets The ide of expoet properties Apply expoet properties Expoets Expoets me repeted multiplictio ( ) Expoet Properties - Multiply Use the ptter to discover the property. Simplify: Expoet Properties 1) x 3 x 7 Scott Eckert pg. 1

2 Expoet Divisio of Sme Bse Use the ptter to discover the property. Simplify: Expoet Properties 1) m = m+ ) x x 11 5 Expoet Zero Power Look t the ptter d drw coclusio Expoet Properties 1) m = m+ m m ) 3) Expoet - Power o Power Use the ptter to discover the property. Simplify: (3 ) 4 (x 3 ) 5 Expoet Properties 1) m = m+ m m ) 3) 0 = 1, for ll except 0. 4) Scott Eckert pg.

3 Expoet Power o Product Use the ptter to discover the property. Simplify: (b) 3 (xy) 5 Expoet Properties 1) r s = r+s ) r s rs 3) 0 = 1, for ll except 0. 4) ( m ) = m 5) Expoet Power o Frctios Use the ptter to discover the property. Expoet Properties Simplify: 1) m = m+ 3 z 4 r rs ) s 3) 0 = 1, for ll except 0. 4) ( m ) = m 5) (b) = b 6) Scott Eckert pg. 3

4 Expoet Prctice Simplify ech b b (t) 8 (t) (-3x) 3 3. ( 3 b)(b) 4 7. ( 4 b 6 )( b) 5 4. x 7 x 8. xy yz 5 5. Negtive Expoets Need To Kow Review Expoets Properties Ide of Negtive Expoets Negtive Expoet Properties d Clcultio Wht is Scietific Nottio? How to write umbers i Scietific Nottio How to do clcultios i Scietific Nottio Scott Eckert pg. 4

5 Review Expoet Properties Recll: The Product Rule m = m+ The Quotiet Rule m m The Power Rule ( m ) = m Risig Product to power Risig quotiet to power (b) = b b b Ide of Negtive Expoets Look the ptter d drw coclusio Defiitios: for ll rel umbers ( 0), Defiitio: for 0 d is positive, Prctice Simplify Ech 5-3 (-) - 5x -4 5 y x z Scott Eckert pg. 5

6 Expoet Properties Expoet of 1 1 = The Product Rule m = m+ Expoet of 0 Negtive Expoets 0 = 1 1 The Quotiet Rule The Power Rule m ( m ) = m m Thik RECIPROCAL Risig Product to power (b) = b Thik RECIPROCAL Risig quotiet to power b b Prctice - Simplify x -6 x (x 4 ) - Prctice - Simplify 5. ( x ) 4 x 3 7. y y x ( x ) ( ) 3 ( ) 5 4 Scott Eckert pg. 6

7 Scietific Nottio Scietific Nottio is wy to write big or smll umbers i compct d simple wy. where N is deciml t lest oe d less th 10 (1 < N < 10) d m is iteger expoet. Exmples of scietific ottio 1) The tiol debt: $ 16,749,09,149, ) The mss of hydroge tom: grms = Scietific Nottio Covertig: Scietific ottio ito expded form x 10 1 = x x 10 = x x 10 5 = x x 10-1 = x x 10-3 = x x x 10 7 Scietific Nottio Covertig: Expded form ito scietific ottio. 35,900, We use the expoet properties to multiply d divide umber i scietific ottio. Exmples: 8 x x 10-3 (7.8 x 10 7 )(8.4 x 10 3 ) Scott Eckert pg. 7

8 5.3 Polyomils Need To Kow Recll like terms Some ew vocbulry Like Terms d polyomils Evlute polyomils Vocbulry RECALL - Defiitios A term is mde of umbers & vribles ofte combied with pretheses, multiplictio or divisio. Like terms re terms with the. A polyomil is fiite sum of terms. Exmples: Moomils Biomils Triomils Other New Vocbulry The degree of term is fctors i the term. (If there is oly oe vrible, the the degree is the expoet.) The degree of polyomil equls where the ledig term is the term i the expressio with the highest degree. The umericl coefficiet is the fctor which multiplies the term. Scott Eckert pg. 8

9 Complete the tble for the polyomil 1w 9 4w w w Terms Coefficiets Degree of Term Ledig Term Degree of Polyomil Polyomils Prctice Whe x = -3 fid the vlue of x x + 3 Recll 3x+ 6x Combie like terms: 7x + x + x 5x 9b 5 + 3b b 5 3b 8x 5 x 4 + x 5 + 7x 4 4x 4 x 6 Scott Eckert pg. 9

10 5.4 Add d Subtrct Polyomils Need To Kow Addig polyomils Opposites of polyomil Subtrctig polyomils Polyomils problems solvig Addig Polyomils (x + 4x 9) + (7x 3) x x 3x 7 x x x Add: x 4 + 3x 3 + 4x 5x 3 6x 3 The Opposite of Polyomil Write the opposite of (x + 3x - 4) i two wys Simplify: ( 5x 6x + 3) 7x 11x x Scott Eckert pg. 10

11 Subtrctig Polyomils Subtrct: (9x + 7) (5x 3) (x + 3x + 4) ( 5x 6x + 3) Subtrct: x + 5x 3 4x 4x 5 Prctice Simplify: (y 7y 8) (6y + 6y 8) + (4y y + 3) Polyomil Problem Solvig Fid the perimeter Fid shded re Scott Eckert pg. 11

12 5.5 Multiplictio of Polyomils Need To Kow Multiply moomil times moomil Multiply moomil times polyomil Multiply polyomil times polyomil Moomil times Moomil Recll Multiplictio: (-x 3 )(x 4 ) Expoet Properties 1) ) (-4y 4 )(6y )(-3y ) 3) Moomil times Polyomil Recll: (b + c) = Expoet Properties 1) m = m+ ) ( m ) = m Multiply: x(4x + 5x - 3) = 3) (b) m = m b m Scott Eckert pg. 1

13 Polyomil times Polyomil Multiply: (x + )(x 3x + 4) Recll Colum Multiply 34 x 13 Polyomil times Polyomil Multiply: colums (z 4)(z + 5) Multiply: (x + x + 1)(x 4x + 3) 5.6 Biomil Multiplictio & Short Cuts Need To Kow Biomils times Biomils Short Cut Product of Sum d Differece Biomil Squres of Biomils Scott Eckert pg. 13

14 Biomil times Biomil Multiply: x + 7 x 5 Multiply: (x + 7)(x 5) Short Cut: FOIL Multiply: F O I L Biomil times Biomil Multiply by distributive lw: (y + 6)(y 3) (3x + 5)(x ) Short Cut: FOIL Multiply: F first terms O outer terms I ier terms L lst terms (x + y)( + 7b) Biomil times Biomil Multiply (x 3)(x 6) Fid the re: (1 + t )(1 3t 3 ) Scott Eckert pg. 14

15 Scott Eckert pg. 15

16 5.7 Multivrible Polyomils Need To Kow Evlutig Polyomil Like Terms d Degree Additio d Subtrctio of Polyomils Multiplictio of Polyomils Evlutig Polyomils A mout of moey P ivested t yerly rte r for t yers will grow to mout of A give by A = P(1 + r) t. Wht will you hve from ivestig $1000 t 6% for 3 yers? New Vocbulry The degree of term is the umber of vrible fctors i the term. The degree of polyomil is the degree of the ledig term, d the ledig term is the term with the highest degree xy 3x y x yz y Terms Coefficiets Degree of Term Ledig Term Degree of Polyomil Scott Eckert pg. 16

17 Simplify: Add d Subtrct Polyomils (x 3xy + y ) + (-4x 6xy y ) + (4x + xy y ) ( 3 + b 3 ) ( b b + 3b 3 ) Multiplyig Polyomils Multiply: (5cd + c d +6)(cd d ) FOILig Polyomils Multiply: (m 3 + 3)(m 3 11) (4r + 3t) (p 3 5q) (p 3 + 5q) ed Scott Eckert pg. 17

18 5.8 Dividig Polyomil Need To Kow Two wys to work divisio Recll the distributive property Divide polyomil by moomil Recll log divisio Divide polyomil by polyomil The Distributive Property Recll: (b + c) = b + c Also: (b + c) = With ew twist: b c (b + c) = Polyomil A B C moo D Divide Polyomil by Moo (5x 10) 5 8x 1x 4x 3 Scott Eckert pg. 18

19 Divide Polyomil by Moo 3 3 (9x y 1 x y ) ( 9 xy) z 14 z 7 z 7z Recll Log Divisio Steps for Divisio Polyomil Divisio x x x 5 6 Steps for Divisio 1. Guess. Multiply 3. Subtrct 4. Brig Dow 5. Repet 8x 6x 5 x 3 Scott Eckert pg. 19

20 Polyomil Divisio 3 t 9t 11t 3 t 3 3 w 10 w Decidig o which wy to DIVIDE Next to ech problem circle the correct wy to divide it. 1. (5x 16 x) (5x 1). 3 (0t 5t 15 t) (5 t) ( ) ( 9 ) x 3x 4x 3 x 5 4x y 8x y 1x y 4 4xy ) Frctio b) Log Divisio ) Frctio b) Log Divisio ) Frctio b) Log Divisio ) Frctio b) Log Divisio ) Frctio b) Log Divisio Scott Eckert pg. 0

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