Math 114- Intermediate Algebra Integral Exponents & Fractional Exponents (10 )

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1 Math 4 Math 4- Itermediate Algebra Itegral Epoets & Fractioal Epoets (0 ) Epoetial Fuctios Epoetial Fuctios ad Graphs I. Epoetial Fuctios The fuctio f ( ) a, where is a real umber, a 0, ad a, is called the epoetial fuctio, base a. Requirig the base to be positive would help to avoid the comple umbers that would occur by takig eve roots of egative umbers. (E., ( ), which is ot a real umber.) The restrictio a is made to eclude the costat fuctio f ( ). Eample: f ( ) 5, f ( ) ( ), 6 f ( ) ( 4. 75) *The variable i a epoetial fuctio is i the epoet. II. Graphig a epoetial fuctio A. Plottig poits B. Graphig calculator * Try f ( ) ( )

2 Math 4 III. Applicatio i Eample: Compoud Iterest Formula, A P( ) r for compoudigs per year: or A P( ) for cotiuous compoudig: A Pe rt. a total of $,000 is ivested at a aual rate of 9%. Fid the balace after 5 years if it is compouded quarterly: r t. ( ) A P( ), 000( ) 8, a total of $,000 is ivested at a aual rate of 9%. Fid the balace after 5 years if it is compouded cotiuously: rt 0. 09( 5) A Pe, 000e 8, cotiuous compoudig yields iterest: =93.64 IV. The Number e A( ) ( ), as the gets larger ad larger, the fuctio value gets closer to e. Its decimal represetatio does ot termiate or repeat; it is irratioal. I 74, Leoard Euler amed this umber e. You ca use the e key o a graphig calculator to fid values of the epoetial fuctio f ( ) e. Eample: Fid e 3 0 3, e., e 0, 00e 5. 8, e t t

3 Math 4 Epoets I. Evaluate epoetial epressios. A. For ay positive iteger, a a a a a a times such that a is the base ad is the epoet Eample: a 3 a a a = 7 B. For ay ozero real umber a ad ay iteger, a 0 Ad a a C. I a multiplcatio problem, the umbers or epressios that are multiplied are called factors. If a b c, the a ad b are factors of c. Eamples: a. 6 0 b. ( ) 0 c. ( ) 3 d. 3 = e. 4 = D. Properties of Epoets a a a m m a a m a ( m) such that a 0 ( a ) ( ab) a m m a b m m m 3

4 Math 4 a ( ) b m m a such that b 0 m b Eamples: a. ( ) 5 b. y 5 y c. y y 6 3 d. ( ) 3 4 e. m 5 5 m a b c f. ( ) a b c 5 = II. Fractioal Epoets Defiitio I.: a a such that is called the ide ad a is the radicad. m Defiitio II: a a ( a ) m m * m is a iteger, is a positive iteger, ad a is a real umber. If is eve, a 0. Eamples: ( ) ( )( ) ( )( ) ( ) ( 5 )

5 Math ( ) ( )( ) Additio ad Subtractio of Polyomials I. Polyomials i Oe Variable a a... a a a 0 ad that is a oegative iteger, ad a,..., a 0 are real umbers called coefficiets ad a 0 A. Defiitio. Terms. Degree of the polyomial 3. Leadig coefficiet 4. Costat term 5. Descedig order Eamples: 4 3 a. 8 0 b. y 6y 3 6. Moomial 5

6 Math 4 7. Biomial 8. Triomial II. Polyomial i Several Variables A. Defiitio. Degree of a term - sum of the epoets of all the variables i that term.. Degree of a polyomial - the degree of the term of highest degree. Eamples: a. 9ab 3 a b 4 9 b. 7 4 y y 3 y 6 III. Epressios That Are Not Polyomials 5 a. 5 b. 0 c. y 3 y 7 6

7 Math 4 IV. Additio ad Subtractio of Polyomials Like terms - terms/epressios that have same variables raised to the same powers. Combie/collect like terms Eamples: 3 3 a. ( 5 3 ) ( 7 3) b. ( 8 y 3 9y) ( 6 y 3 3y) c. ( 3 3 ) ( ) 7

8 Math 4 Multiplicatio of Polyomials A. (biomial)(biomial) : biomial multiplied by biomial, use FOIL ( 4)( 3 ) ( ) ( 3) ( 4 ) ( 4 3) F O I L 3 4 = 7 Eamples: a. ( 5)( 3) b. ( a 3)( a 5) c. ( 3y)( 5y) = d. ( 4 ) ( 4 )( 4 ) e. ( 5 ) ( 5 )( 5 ) 8

9 Math 4 f. ( 3y ) ( 3y )( 3y ) g. ( a ) ( a )( a ) h. ( a )( a ) i. ( 3y )( 3y ) j. ( 5 )( 5 ) B. Special Products of Biomials. ( A B) A AB B. ( A B) A AB B 3. ( A B)( A B) A B C. Multiplyig Two Polyomials Eamples: a. ( a b)( a 3 ab 3b ) b. ( 4 4 y 7 y 3y)( y 3 y) 9

10 Math 4 c. ( 3y 4)( y) Divisio of Polyomials ad Sythetic Divisio I. Divide a Polyomial by a Moomial Divide each term of the polyomial by the moomial A. Eample: B. Try: y 6 y 3 y 5 y y 6 y 3 y 3 y abc 5a b c 8ab c 3 3ab c yz 6yz 9 y z 6y a b c 6ab c 5a b abc 3 5 0

11 Math 4 II. Divide a Polyomial by a Biomial Divide a ppolyomial by a biomial as we perform log divisio. A. Eamples:

12 Math 4 III. Divide Polyomials Usig Sythetic Divisio A. Whe a polyomial is divided by a biomial of the form a, the divisio process ca be greatly shorteed by sythetic divisio. Eamples

13 Math 4 *Please check agaist results from.,., ad 4. from previous page. B. Polyomial Divisio. Factor: Whe we are dividig oe polyomial by aother, we obtai a quotiet ad a remaider. If the remaider is 0, the the divisor is a factor of the divided. P( ) d( ) Q( ) R( ) P( ) is the divided, d( ) is the divisor, Q( ) is the quotiet, R( ) is the remaider Eample: Determie 3 a. Whether is a factor of 5 6 b. Whether 3 is a factor of 4 8 3

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