MATH 115: Review for Chapter 5

Transcription

1 MATH 5: Review for Chapter 5 Can you find the real zeros of a polynomial function and identify the behavior of the graph of the function at its zeros? For each polynomial function, identify the zeros of the function and the multiplicity of each zero. Determine whether the graph of the function crosses or touches the -ais at each -intercept. Sketch the graph. () f ( ) 4 ( ) ( ) g = 4 + = + + () ( ) ( ) ( ) 4 () f ( ) = + ( ) (4) g ( ) = ( ) ( 9) = + + (6) g ( ) = ( ) ( ) ( + ) (5) f ( ) ( ) ( ) ( ) Can you solve a problem using the Cubic Regression feature of a graphing utility? Solve each application problem using a graphing utility. (7) The following data represents the sales (in millions of dollars) of a product for the years , where represents 990. Use a graphing calculator to draw a scatter diagram of the data and to find the cubic function of best fit. Use that function to predict the amount of sales in 999 to the nearest integer. Year Sales S $8 $7 $4 4 $47 5 $5 6 $55 7 $6 8 $7 9 $86 (8) The following data represents the dollar cost per stack of plywood in construction for the years , where represents 990. Use a graphing calculator to draw a scatter diagram of the data and to find the cubic function of best fit. Use that function to predict the cost in 999 to the nearest dollar. Year Cost C $ $0 $5 4 $8 5 $4 6 $49 7 $59 8 $7 9 $94 Revised 0/7/07

2 Can you eplain how to use algebra and sign charts to solve polynomial and rational inequalities? Can you solve polynomial inequalities algebraically and verify your results graphically? Solve the inequalities. (9) + 6 (0) () 9 0 > () 4 < 0 Can you solve rational inequalities algebraically and verify your results graphically? Solve the inequalities. () 7 5 (4) > 0 (5) 0 Can you state the Factor Theorem for polynomials? Can you use the Factor Theorem to test factors of a polynomial? Use the Factor Theorem to determine if the indicated factor is a factor of the given polynomial. (6) Is a factor of (7) Is + a factor of (8) Is + a factor of ? ? ? Can you use a linear factor of a polynomial and synthetic (or long) division to write a polynomial in factored form? (9) Check that + is a factor of ( ) factorization for f ( ). (0) Check that is a factor of ( ) factorization for f ( ). () Check that is a factor of ( ) factorization for f ( ). 4 = 0 + and then provide a = and then provide a = and then provide a Can you state the Rational Zeros Theorem for polynomials with integer coefficients? Can you use the Rational Zeros Theorem to list the potential rational zeros for a polynomial function? List the potential rational zeros of the polynomial function and use your grapher to determine which of the possibilities are actual zeros. () ( ) = () ( ) f = Revised 0/7/07

3 (4) ( ) f = Can you use the Rational Zeros Theorem and synthetic division in conjunction with a graphing utility to find the real zeros of a polynomial function? Find the real zeros of the polynomial function and use the zeros to factor the function. (5) ( ) (7) ( ) = + 4 (6) ( ) = (8) ( ) Can you find all comple zeros of a polynomial function? f = f = Find the comple zeros of the polynomial function and use them to factor the function completely in the comple system. (9) ( ) () ( ) 5 4 = (0) f ( ) = = Can you construct a polynomial function with specified real or comple zeros? Write a polynomial function with real coefficients that has the given comple zeros. (), 5i () 4 i, (4) i, 6i (5) 7, 7, i Revised 0/7/07

4 Answers: (), multiplicity, crosses -ais, multiplicity, crosses -ais (), multiplicity, crosses -ais, multiplicity, crosses -ais, multiplicity, touches -ais (), multiplicity, crosses -ais 4, multiplicity, crosses -ais (4), multiplicity, crosses -ais, multiplicity, touches -ais (5), multiplicity, touches -ais, multiplicity, crosses -ais, multiplicity, crosses -ais (6), multiplicity, crosses -ais, multiplicity, crosses -ais, multiplicity, crosses -ais, multiplicity, touches -ais (7) S ( ) Sales in 999 ( = 0) are approimately $06 million. (8) C ( ) Cost in 999 ( = 0) is approimately $. (9) Then + 6 = 0 ( + ) ( ) = 0 = or =, so we form the sign chart: Interval f ( ) (, ) positive (, ) = + 6 negative (, ) positive so the solution is [, ] (0) (, 5] [, ) Graphical solution for (0): (),0, () (,0) ( 0, ) Revised 0/7/07

5 () ( ) Then + 8 = 0 = 8 and 5 = 0 = 5, so we form the 5 sign chart: Interval f ( ) (,5) negative ( 5, 8 ) positive ( 8, ) negative = so the solution is (,5) [ 8, ) (4) (, 5) [, ) Graphical solution for (4): (5) 0 0. Then, from the numerator, = 0 = ± and from the denominator we have = 0, so we form the sign chart: (6) ( ) = positive Interval f ( ) (, ) (, 0) negative ( 0, ) positive (, ) negative The solution is [, 0) [, ). f = = 0, so (7) f ( ) = 0 so + is a factor. (8) f ( ) = 9 so + is not a factor. (9) Using synthetic division, ( ) ( ) ( ) f = , so 0 is a factor. (0) Using synthetic division, , so f ( ) = ( + ) Revised 0/7/07

6 () Using synthetic division, , so f ( ) ( ) ( 0 5 6) = +. () () ±, ±, ±, ± 4 ; actual zero: ±, ±, ±, ±, ±, ±, ±, ±, ±, ± 5, ±, ± 5; actual zeros: ,, (4) ±, ±, ± 5, ± 9, ± 5, ± 45, ±, ±, ±, ±, ±, ± ; actual zeros:,,,5 (5) zeros:,, ; factorization: f ( ) = ( ) ( ) = ( + ) ( ) = ( + ) ( ) ( ) (6) solution: possible rational zeros are: ±, ±, ±, ±, ±, ±, ±, ±, ±, ± 5, ±, ± 5. The graph indicates that = is a good candidate for a zero. Using synthetic division: , so = is a zero and the quadratic formula on ( ) ( ) ( ) ( ) ( ) ( ) to obtain 6 ± ± 4 = = so that is a factor. Use 5 = or =. 5 So the zeros are,, and the factorization is 5 5 f ( ) = 4 ( ) = ( + ) = ( + ) ( ) ( 5). Revised 0/7/07

7 5 (7) zeros:, 5 + 7, 5 7 ; factorization: = (8) zeros: ( 5) ( 5 ) = + + 6,, (multiplicity ); ( ) factorization: f ( ) = ( ) ( ) ( 6) = ( ) ( + 6) ( ) ( ) ( 6) = + (9) zeros: (multiplicity ),, i, i ; f = + i + i factorization: ( ) ( ) ( ) ( ) ( ) ( ) (0) solution: possible rational zeros are ±, ±, ±, ± 4, ± 5, ± 6, ± 0, ±, ± 5, ± 0, ± 0, ± 60. The graph indicates that = and = are good candidates for zeros. Using synthetic division: , so = is a zero and is a factor. Using =, , so = is a zero and + 0 is a factor. 0 0 Apply the quadratic formula to + 0 to obtain ± 4 40 ± 6i = = ± i. So the zeros are,, + i, i, and the factorization is f ( ) = ( ) ( + ) ( + i) ( i) = + i + i. ( ) ( ) ( ) ( ) Revised 0/7/07

8 () zeros: 5, 4 + i, 4 i ; factorization: f ( ) = ( + 5) ( 4 + i) ( 4 i) = i i ( ) ( ) ( ) For problems () through (5), answers may vary, but these are obvious solutions. () Using the conjugate pairs theorem, ( ) ( ) ( ) ( ) ( ) ( ) f = i i = + = () f ( ) = ( 4 i) ( 4 + i) ( ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( ) = + i i + + i i + ( ) ( ) = + + = (4) ( ) ( ) ( ) ( ) ( ) ( ) ( ) f = i i i i = + + = (5) ( ) ( ) ( ) ( ) ( ) ( ) ( ) f = i i = + = Revised 0/7/07