Exponents, Polynomials, and Polynomial Functions

Transcription

1 Eponents, Polynomials, and Polynomial Functions. Integer Eponents R. Fractions and Scientific Notation Objectives. Use the product rule for eponents.. Define and negative eponents.. Use the quotient rule for eponents.. Use the power rule for eponents.. Simplify eponential epressions. 6. Use the rules for eponents with scientific notation. Slide Section of., Slide Section of., Use the product rule for eponents. The products of eponential epressions with the same base are found by adding eponents. Eample Using as an Eponent Apply the product rule for eponents, if possible, in each case. a. 6 b. 6 6 Slide Section of., c. p p p ( )()p p p p 8 p d. a b The bases are not the same, so the product rule does not apply. Slide Section of., Define and negative eponents Eample Evaluate. a. b. ( ) Using as an Eponent c. ( ) d. ( ) Slide Section of., e. f. 6 (8 y) Any nonzero quantity raised to the zero power equals. Slide 6 Section of., 6

2 Negative Eponents Eample Using Negative Eponents Write with only positive eponents. a. b. c. p ( p) d. 6 z 6 z 6z e. a ( a) ( a) a f. Slide Section of., Slide 8 Section of., 8 Special Rules for Negative Eponents Use the quotient rule for eponents. Slide 9 Section of., 9 Slide Section of., Eample Using the Quotient Rule for Eponents Apply the quotient rule for eponents, if possible, write each result using only positive eponents. a. Use the power rule for eponents. b. w w w ( ) w w c. a b The quotient rule does not apply because the bases are different. Slide Section of., Slide Section of.,

3 Use the power rule for eponents. Eample 6 Simplify, using the power rules. More Special Rules for Negative Eponents Slide Section of., Slide Section of., Eample Using Negative Eponents with Fractions Write each epression with positive eponents and then evaluate. a. b Using the Definitions and Rules for Eponents Eample 8 Simplify each epression so that no negative eponents appear in the final result. Assume that all variables represent nonzero real numbers. a. ( ) c. y z k k y z 9k y z 6 b. y z yz Slide Section of., Slide 6 Section of., 6 Scientific Notation Scientists often deal with etremely large and etremely small numbers. For eample: Scientific Notation It is often simpler to write these very large or very small numbers using scientific notation. The distance from the sun to the Earth is approimately,, kilometers. X-ray The wavelength of -rays is. meter. Slide Section of., Slide 8 Section of., 8

4 Converting to Scientific Notation To write numbers in scientific notation, we use the following steps. Eample 9 Writing Numbers in Scientific Notation Write each number in scientific notation. a.,. 8 6 Move the decimal point 8 places to the right. 8. b If the number is negative, use the steps above and then label your results as a negative number. Slide 9 Section of., 9 Move the decimal point 6 places to the left. 6.8 Slide Section of., Converting from Scientific Notation to Standard Notation Eample Write each number in standard notation. a. 6.9 b. Move the decimal point places to the right. 69,,. 6 Move the decimal point 6 places to the left.. Using Scientific Notation to Solve Problems Eample In 99, the national health care ependiture in the United States was $. billion. By 9, this figure had risen by a factor of. that is, it more than tripled in about yr. a. Write the 99 health care ependiture using scientific notation.. billion 9. 9 (. ). Slide Section of., Slide Section of., Using Scientific Notation to Solve Problems Eample In 99, the national health care ependiture in the United States was $. billion. By 9, this figure had risen by a factor of. that is, it more than tripled in about yr. b. What was the ependiture in 9? Multiply the result in part (a) by.. (. ). (..).99 (.99 ).99 Slide Section of.,. Adding and Subtracting R. Fractions Polynomials Objectives. Know the basic definitions for polynomials.. Add and subtract polynomials. Slide Section of.,

5 Know the basic definitions for polynomials. Polynomials are fundamental in algebra. Recall that a term is a number, a variable, or the product or quotient of a number of one or more variables raised to powers. Eamples of terms include: Know the basic definitions for polynomials. Recall that any combination of variables or constants (numerical values) joined by the basic operations of addition, subtraction, multiplication, and division (ecept by ), or raising to powers or taking roots is called an algebraic epression. Polynomials are the simplest kind of algebraic epression. Coefficients are written in blue. Slide Section of., Slide 6 Section of., 6 Know the basic definitions for polynomials. A polynomial containing only the variable is called a polynomial in. A polynomial in one variable is written in descending powers of the variable if the eponents on the variable decrease from left to right. The Degree of a Polynomial The degree of a nonzero term with only one variable is the eponent on the variable. The number has no degree. The degree of a polynomial is the highest degree of any of its nonzero terms. The powers of are decreasing from left to right. We can think of this polynomial as Slide Section of., Slide 8 Section of., 8 Some Common Polynomials Polynomials having a specific number of terms are commonly given special names. Trinomial = a polynomial with three terms Binomial = a polynomial with two terms Monomial = a polynomial with one term Eample Classifying Polynomials Identify each polynomial as a monomial, a binomial, a trinomial, or none of these. Also, give the degree. a. b. y Trinomial; degree. Monomial; degree ( + = ) c. p p p 6 None of these; degree. Slide 9 Section of., 9 Slide Section of.,

6 Adding Polynomials Add and Subtract Polynomials Addition of polynomials is just a matter of adding up like terms. For eample, consider the following polynomials: We can use the associative and commutative properties to rearrange the terms Eample Add: Add: Vertical Solution and then we add the like terms. Slide Section of., Slide Section of., Subtracting Polynomials Eample Subtract. Subtracting Polynomials Slide Section of., Slide Section of., Subtracting Polynomials Eample We can subtract these polynomials vertically by writing the first polynomial above the second, lining up like terms in columns.. Polynomial Functions R. Fractions Objectives. Recognize and evaluate polynomial functions.. Use a polynomial function to model data. Change all the signs in the second polynomial, and add.. Add and subtract polynomial functions.. Graph basic polynomial functions. Slide Section of., Slide 6 Section of., 6 6

7 Definition of a Polynomial Function Eample Evaluating Polynomial Functions Polynomial Function Let f() = +. Find each value. A polynomial function of degree n is defined by f () = a n n + a n n + + a + a, for real numbers a n,a n,..., a, and a, where a n and n is a whole number. (a) f() f() = + f() = + = 8 + = + = 9 Slide Section of., Slide 8 Section of., 8 Evaluating Polynomial Functions Continued. Let f() = +. Find each value. (b) f( ) Functions While f is the most common letter used to represent functions, recall that other letters such as g and h are also used. The capital letter P is often used for polynomial functions. f() = + f( ) = ( ) ( ) + = ( ) 9 + = 8 + = 6 Slide 9 Section of., 9 Slide Section of., Using a Polynomial Model to Approimate Data Eample The number of U.S. households estimated to see and pay at least one bill on-line each month during the years through 6 can be modeled by the polynomial function defined by P() = , where = corresponds to the year, = corresponds to, and so on, and P() is in millions. Use this function to approimate the number of households epected to pay at least one bill on-line each month in 6. Since = 6 corresponds to 6, we must find P(6). P() = P(6) =.88(6) +.6(6) +. Let = 6. =. Evaluate. Thus, in 6 about. million households are epected to pay at least one bill on-line each month. Slide Section of., Adding and Subtracting Functions Adding and Subtracting Functions If f() and g() define functions, then (f + g) () = f () + g() Sum function and (f g) () = f () g(). Difference function In each case, the domain of the new function is the intersection of the domains of f() and g(). Slide Section of.,

8 Eample Adding and Subtracting Functions For the polynomial functions defined by f() = + and g() = + 9, find (a) the sum and (b) the difference. (a) (f + g) () = f () + g() Use the definition. = ( + ) + ( + 9 ) Substitute. = + 6 Add the polynomials. (b) (f g) () = f () g() Use the definition. = ( + ) ( + 9 ) Substitute. = ( + ) + ( 9 + ) Change subtraction to addition. = + 9 Add. Slide Section of., Eample Adding and Subtracting Functions For the polynomial functions defined by f() = and g() =, find each of the following. (a) (f + g) () (f + g) () = f () + g() Use the definition. = [() ] + () Substitute. = Slide Section of., Eample Adding and Subtracting Functions For the polynomial functions defined by f() = and g() =, find each of the following. (a) (f + g) () Alternatively, we could first find (f + g) (). (f + g) () = f () + g() Use the definition. = ( ) + Substitute. = + Then, (f + g) () = () + () =. The result is the same. Slide Section of., Eample Adding and Subtracting Functions For the polynomial functions defined by f() = and g() =, find each of the following. (b) (f g) () and (f g) () Then, (f g) () = f () g() Use the definition. = ( ) Substitute. = Combine like terms. (f g) () = () () = Substitute. Confirm that f () g() gives the same result. Slide 6 Section of., 6 Basic Polynomial Functions Basic Polynomial Functions The simplest polynomial function is the identity function, defined by f() =. y The squaring function, is defined by f() =. y f() = f() = Slide Section of., Slide 8 Section of., 8 8

9 Basic Polynomial Functions Graphing Variations of Polynomial Functions The cubing function, is defined by f() =. f() = 8 8 y Eample Graph the function by creating a table of ordered pairs. Give the domain and the range of the function by observing the graph. (a) f() =. y f() = Range Domain Slide 9 Section of., 9 Slide Section of., Graphing Variations of Polynomial Functions Continued. Graph the function by creating a table of ordered pairs. Give the domain and the range of the function by observing the graph. (b) f() =. y f() = Range Domain. Multiplying Polynomials R. Fractions Objectives. Multiply terms.. Multiply any two polynomials.. Multiply binomials.. Find the product of the sum and difference of two terms.. Find the square of a binomial. 6. Multiply polynomial functions. Slide Section of., Slide Section of., Eample Multiply Terms Eample Multiplying Polynomials a b. a b ab ()a b ab a b 6 a b (a) ( 6 + ) = ( 6 ) + ( ) = 6 + Distributive property Multiply monomials. Slide Section of., Slide Section of., 9

10 Continued. Multiplying Polynomials Continued. Multiplying Polynomials (b) h ( h 9 + 8h ) = h ( h 9 ) = 6h 6h 6 + h + ( h ) (8h ) Distributive property + ( h )( ) Multiply monomials. Slide Section of., ( y ) ( y y + ) = (y )(y ) + (y )( y) + (y )() + ( ) (y ) + ( )( y) + ( )() = y y + 8y y + y Distributive property = y y + 8y + y Combine like terms. Slide 6 Section of., 6 Multiplying Polynomials Vertically Eample Find the product. Multiply n n 8n n vertically. n + 8n n n + n + n 9n 6 n + n n 8n n + n + n + 6n 9n 6 Multiply Binomials by the FOIL Method Eample Use the FOIL method to find y y. ( y ) ( y ) Multiply the first terms: Multiply the outer terms: Multiply the inner terms: F O I L = y y y + 8 = y y + 8 y ( y ) y ( ) ( y ) F O I Multiply the last terms: ( ) L Slide Section of., Slide 8 Section of., 8 Multiply Binomials by the FOIL Method Eample Find the product g 9g. ( g + ) ( 9g ) Multiply the first terms: F O I L = g g + 8g 8 = g + 6g 8 g ( 9g ) F Multiply Binomials by the FOIL Method Eample Find the product 6a b a b. ( 6a + b ) ( a b ) Multiply the first terms: F O I L = a ab + ab 6b = a 6b 6a ( a ) F Multiply the outer terms: g ( ) O Multiply the outer terms: 6a ( b ) O Multiply the inner terms: ( 9g ) I Multiply the inner terms: b ( a ) I Multiply the last terms: ( ) L Multiply the last terms: b ( b ) L Slide 9 Section of., 9 Slide 6 Section of., 6

11 Product of the Sum and Difference of Two Terms Multiplying the Sum and Difference of Two Terms Eample (a) (m + 8n)(m 8n) = (m) (8n) = 9m 6n (b) p(p + )(p ) = p[(p) () ] = p(p ) = p p Slide 6 Section of., 6 Slide 6 Section of., 6 Squaring Binomials Finding the Square of a Binomial Eample 6 a. (m + ) m m m m 9 c. ( p v ) p ( p)( v) ( v) p pv 9v b. (p ) p p p p Slide 6 Section of., 6 Slide 6 Section of., 6 Eample Multiplying More Complicated Polynomials Eample Multiplying More Complicated Polynomials a. ( s) = ( s)( s) = ( s)[ ()(s) + (s) ] = ( s)(9 6s + 6s ) = 9s + s 96s + s 6s = 88s + 6s 6s (b) (t + v) = (t + v) (t + v) = [(t) + (t)(v) + (v) ][(t) + (t)(v) + (v) ] = (t + tv + 9v )(t + tv + 9v ) = 6t + 8t v + 6t v + 8t v + t v + 8tv + 6t v + 8tv + 8v = 6t + 96t v + 6t v + 6tv + 8v Slide 6 Section of., 6 Slide 66 Section of., 66

12 Multiplying Polynomial Functions Multiplying Polynomial Functions In Section., we saw how functions could be added and subtracted. Functions can also be multiplied. Eample 8 For f() = and g() = + find (fg)() and (fg)( ). ( fg)( ) f ( ) g( ) ( fg)( ) ( )( ) 9 Slide 6 Section of., 6 ( fg)( ) ( ) ( ) ( ) 6 Slide 68 Section of., 68. Dividing Polynomials R. Fractions Objectives Dividing a Polynomial by a Monomial. Divide a polynomial by a monomial.. Divide a polynomial by a polynomial of two or more terms.. Divide polynomial functions. Slide 69 Section of., 69 Slide Section of., Dividing a Polynomial by a Monomial Eample 8 6 Divide. Dividing a Polynomial by a Monomial Eample Divide. 8y y 6y y We can check this answer by multiplying the quotient by the divisor to obtain the original polynomial. Slide Section of., Why is the answer not a polynomial? y is in the denominator All variables must have whole number eponents and no variables can be in the denominator or under radicals. Slide Section of.,

13 Eample Dividing a Polynomial by a Polynomial Dividing a Polynomial by a Polynomial Slide Section of., Slide Section of., Dividing a Polynomial by a Polynomial Eample Dividing a Polynomial with a Missing Term Slide Section of., Slide 6 Section of., 6 Dividing a Polynomial with a Missing Term Dividing a Polynomial with a Missing Term Slide Section of., Slide 8 Section of., 8

14 Dividing a Polynomial with a Missing Term Dividing a Polynomial with a Missing Term Slide 9 Section of., 9 Slide 8 Section of., 8 Dividing a Polynomial with a Missing Term Eample Dividing a Polynomial with a Missing Term Slide 8 Section of., 8 Slide 8 Section of., 8 Dividing Polynomial Functions Dividing Polynomial Functions Eample 6 continued on net slide Slide 8 Section of., 8 Slide 8 Section of., 8

15 Eample 6 Dividing Polynomial Functions Applying the definition of dividing functions: Performing division as shown earlier Results are the same Slide 8 Section of., 8