171S4.1p Polynomial Functions and Models
MAT 171 Precalculus Algebra
Dr. Claude Moore
Cape Fear Community College
CHAPTER 4:
Polynomial and Rational Functions
4.1 Polynomial Functions and Models
4.2 Graphing Polynomial Functions
4.3 Polynomial Division; The Remainder and Factor Theorems
4.4 Theorems about Zeros of Polynomial Functions
4.5 Rational Functions
4.6 Polynomial and Rational Inequalities
4.1 Polynomial Functions and Models
•
Determine the behavior of the graph of a polynomial function using the leadingterm test.
Factor polynomial functions and find the zeros and their multiplicities.
•
Use a graphing calculator to graph a polynomial function and find its realnumber zeros its relative maximum and minimum values, and its domain and range.
•
Solve applied problems using polynomial models; fit linear, quadratic, power, cubic, and quartic polynomial functions to data.
The following PowerPoint presentation was developed by Dr. Moore.
It covers material in sections as indicated below.
Section 4.1, Slides 712 http://cfcc.edu/faculty/cmoore/c2s14.ppt
Section 4.2, Slides 1314 http://cfcc.edu/faculty/cmoore/c2s14.ppt
Section 4.3, Slides 1520 http://cfcc.edu/faculty/cmoore/c2s14.ppt
Section 4.4, Slides 2126 http://cfcc.edu/faculty/cmoore/c2s14.ppt
Oct 1812:48 PM
Oct 238:07 AM
Polynomial Function
A polynomial function P is given by where the coefficients a n , a n 1
, …, a
1
, a
0
are real numbers and the exponents are whole numbers.
Quadratic Function
October 28, 2012
Oct 237:55 AM
Technology for this section
Click the globe to the left and visit SAS Curriculum
Pathways for interactive programs on Polynomial
Functions.
User: able7oxygen
Quick Launch: 1022 (Worksheet: Polynomial Patterns), 1441
(Exploring Graphs of Polynomial Functions)
You may use the "Polynomial Roots" program to graph polynomial functions and find the real roots (zeros). http://cfcc.edu/mathlab/geogebra/poly_roots.html
Mathematica Interactive Figures are available through
Tools for Success, Activities and Projects in CourseCompass.
You may access these through CourseCompass or from the
Important Links webpage. You must Login to MML to use this link.
4.1 Polynomials and the LeadingTerm Test; 4.1 Zeros of Polynomial Functions
4.1 Volume of a Box; 4.1 Modeling Data: Regression
Oct 1812:48 PM
Cubic Function
Examples of Polynomial Functions
Oct 1812:48 PM Oct 1812:48 PM
1
171S4.1p Polynomial Functions and Models
The LeadingTerm Test
October 28, 2012
Oct 1812:48 PM
Graphs
Solution
Oct 1812:48 PM
Find the zeros of f ( x ) = 0.2
x 3 – 1.5
x 2 – 0.3x + 2.
Approximate the zeros to three decimal places.
Solution Use a graphing calculator to create a graph. Look for points where the graph crosses the x axis. We use the ZERO feature to find them.
The zeros are approximately –1.164, 1,142, and 7.523. http://cfcc.edu/mathlab/geogebra/poly_roots.html
Oct 1812:48 PM
a)
b)
c)
d)
Example
Using the leading termtest, match each of the following functions with one of the graphs
AD, which follow.
Oct 1812:48 PM
Finding Zeros of Factored Polynomial Functions
c
f
c
c
x
Finding Zeros of Factored Polynomial
Functions continued
Solution:
To solve the equation f(x) = 0, we use the principle of zero products, solving x 1 = 0 and x + 2 = 0.
The zeros of f(x) are 1 and 2.
See graph on right.
Oct 1812:48 PM
If ( x
−
c ) k , k
≥
1, is a factor of a polynomial function P ( x ) and
( x
−
c ) k + 1 is not a factor and:
•
k is odd, then the graph crosses the x axis at ( c , 0);
•
k is even, then the graph is tangent to the x axis at ( c , 0).
f
x
x 4
x 2
f
x
x 4
x 2
x 2
x 2
f
x
Oct 1812:48 PM
2
171S4.1p Polynomial Functions and Models
309/2.
Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial function as constant, linear, quadratic, cubic, or quartic.
f(x) = 15x 2 10 + 0.11x
4 7x 3
October 28, 2012
309/8.
Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial function as constant, linear, quadratic, cubic, or quartic.
f(x) = 2 x 2
309/5.
Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial function as constant, linear, quadratic, cubic, or quartic.
f(x) = 305x 4 + 4021
Oct 182:19 PM
309/12.
Select one of the following four sketches to describe the end behavior of the graph of the function.
f(x) = (1/4)x 4 + (1/2)x 3 6x 2 + x 5
309/14.
Select one of the following four sketches to describe the end behavior of the graph of the function.
f(x) = (2/5)x 5 2x 4 + x 3 (1/2)x + 3
Oct 182:19 PM
310/19.
Use the leadingterm test to match the function with one of the graphs (a) (d), which follow. f(x) = x 6 + 2x 5 7x 2
309/10.
Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial function as constant, linear, quadratic, cubic, or quartic.
f(x) = 12 + x
Oct 182:19 PM
309/16.
Select one of the following four sketches to describe the end behavior of the graph of the function.
f(x) = x 3 + x 5 0.5x
6
309/18.
Select one of the following four sketches to describe the end behavior of the graph of the function.
f(x) = 2x + x 3 5x 5
Oct 182:19 PM
310/21.
Use the leadingterm test to match the function with one of the graphs (a) (d), which follow. f(x) = x 5 + (1/10)x 3
310/20.
Use the leadingterm test to match the function with one of the graphs (a) (d), which follow. f(x) = 2x 4 x 2 + 1
Oct 182:19 PM
310/22.
Use the leadingterm test to match the function with one of the graphs (a) (d), which follow. f(x) = x 3 + x 2 2x + 4
Oct 182:19 PM
3
171S4.1p Polynomial Functions and Models
310/23.
Use substitution to determine whether 4, 5, and 2 are zeros of f(x) = x 3 9x 2 + 14x + 24
October 28, 2012
310/25.
Use substitution to determine whether 2, 3, and 1 are zeros of f(x) = x 4 6x 3 + 8x 2 + 6x 9 f(5) = (5) 3 9(5) 2 + 14(5) + 24 = 125 9(25) + 70 + 24 = 6
Thus, x = 5 is NOT a zero of f(x).
f(2) = (2) 3 9(2) 2 + 14(2) + 24 = 8 9(4) 28 + 24 = 48
Thus, x = 2 is NOT a zero of f(x).
310/24.
Use substitution to determine whether 2, 3, and 1 are zeros of f(x) = 2x 3 3x 2 + x + 6
310/26.
Use substitution to determine whether 1, 2, and 3 are zeros of f(x) = x 4 x 3 3x 2 + 5x 2
Oct 182:19 PM
310/28.
Find the zeros of the polynomial function and state the multiplicity of each
f(x) = (x + 5) 3 (x 4)(x + 1) 2
Oct 182:19 PM
310/38.
Find the zeros of the polynomial function and state the multiplicity of each
f(x) = x 4 10x 2 + 9
310/34.
Find the zeros of the polynomial function and state the multiplicity of each
f(x) = x 2 (x + 3) 2 (x 4)(x + 1) 4
310/42.
Find the zeros of the polynomial function and state the multiplicity of each
f(x) = 3x 3 + x 2 48x 16
Oct 182:19 PM
310/44.
Using a graphing calculator, find the real zeros of the function f(x) = x 3 + 3x 2 9x 13
You may wish to use this program to check your work. http://cfcc.edu/mathlab/geogebra/poly_roots.html
310/46.
Using a graphing calculator, find the real zeros of the function f(x) = x 4 2x 3 5.6
You may wish to use this program to check your work. http://cfcc.edu/mathlab/geogebra/poly_roots.html
Oct 182:19 PM
Oct 182:19 PM
310/48.
Using a graphing calculator, find the real zeros of the function f(x) = 3x 3 x 2 14x 10
You may wish to use this program to check your work. http://cfcc.edu/mathlab/geogebra/poly_roots.html
Oct 182:19 PM
4
171S4.1p Polynomial Functions and Models
310/50.
Using a graphing calculator, find the real zeros of the function f(x) = x 6 10x 5 + 13x 3 4x 2 5
You may wish to use this program to check your work. http://cfcc.edu/mathlab/geogebra/poly_roots.html
October 28, 2012
310/52.
Using a graphing calculator, estimate the real zeros, the relative maxima and minima, and the range of the function
h(x) = (1/2)x 4 + 3x 3 5x 2 + 3x + 6
310/54.
Using a graphing calculator, estimate the real zeros, the relative maxima and minima, and the range of the function
h(x) = 2x 3 x 4 + 20
Oct 182:19 PM Oct 182:19 PM
310/55.
Using a graphing calculator, estimate the real zeros, the relative maxima and minima, and the range of the function
f(x) = x 2 + 10x + x 5
310/56.
Using a graphing calculator, estimate the real zeros, the relative maxima and minima, and the range of the function
f(x) = 2x 4 5.6x
2 + 10
Oct 182:19 PM
311/57.
Determine whether true or false:
If P(x) = (x 3) 4 (x + 1) 3 , then the graph of the polynomial function y = P(x) crosses the xaxis at (3, 0).
311/58.
Determine whether true or false:
If P(x) = (x + 2) 2 (x 1/4) 5 , then the graph of the polynomial function y = P(x) crosses the xaxis at
(1/4, 0).
Oct 182:19 PM
Oct 182:19 PM
311/59.
Determine whether true or false:
If P(x) = (x 2) 3 (x + 5) 6 , then the graph of the polynomial function y = P(x) crosses the xaxis at
(1/4, 0).
311/60.
Determine whether true or false:
If P(x) = (x + 4) 2 (x 1) 2 , then the graph of the polynomial function y = P(x) crosses the xaxis at
(1/4, 0).
Oct 182:19 PM
5
171S4.1p Polynomial Functions and Models
311/64.
Threshold Weight . In a study performed by
Alvin Shemesh, it was found that the threshold weight W , defined as the weight above which the risk of death rises dramatically, is given by
W(h) = (h/12.3) 3 , where W is in pounds and h is a persons height, in inches. Find the threshold weight of a person who is 5 ft 7 in. tall.
October 28, 2012
W(h) = (h / 12.3)3 Substitute 5 ft 7 in. = 67 in. into the formula for h = 67. Thus, W(67) = (67 / 12.3)3 = 161.625 lbs.
The threshold weight for a person 5 ft 7 in. is about 161.6 lbs.
Oct 182:19 PM
311/65.
Projectile Motion.
A stone thrown downward with an initial velocity of 34.3 m/sec will travel a distance of s meters, where s(t) = 4.9t
2 + 34.3t and t is in seconds. If a stone is thrown downward at
34.3 m/sec from a height of 294 m, how long will it take the stone to hit the ground?
Oct 258:00 AM
311/??.
Windmill Power.
Under certain conditions, the power P, in watts per hour, generated by a windmill with winds blowing v miles per hour is given by P(v) = 0.015v
3 .
(a) Find the power generated by 15mph winds.
(b) How fast must the wind blow in order to generate
120 watts of power in 1 hr?
Oct 182:19 PM
312/70. Interest Compounded Annually. When P dollars is invested at interest rate i , compounded annually, for t years, the investment grows to A dollars, where A = P(1 + i) t . When Sara enters the 11th grade, her grandparents deposit $10,000 in a college savings account. Find the interest rate i if the $10,000 grows to
$11,193.64 in 2 years.
Oct 182:19 PM
Oct 182:19 PM
For instructions of using TI to model, see the tutorial at http://cfcc.edu/faculty/cmoore/TI83Modeling.htm
313/72.
Determine which, if any, of the following functions might be used as a model for the data. a) Linear, f(x) = mx + b b) Quadratic, f(x) = ax 2 + bx + c, a > 0 c) Quadratic, f(x) = ax 2 + bx + c, a < 0 d) Polynomial, not linear or quadratic
313/73.
Determine which, if any, of the following functions might be used as a model for the data. a) Linear, f(x) = mx + b b) Quadratic, f(x) = ax 2 + bx + c, a > 0 c) Quadratic, f(x) = ax 2 + bx + c, a < 0 d) Polynomial, not linear or quadratic
313/74.
Determine which, if any, of the following functions might be used as a model for the data. a) Linear, f(x) = mx + b b) Quadratic, f(x) = ax 2 + bx + c, a > 0 c) Quadratic, f(x) = ax 2 + bx + c, a < 0 d) Polynomial, not linear or quadratic
313/76. Determine which, if any, of the following functions might be used as a model for the data. a) Linear, f(x) = mx + b b) Quadratic, f(x) = ax 2 + bx + c, a > 0 c) Quadratic, f(x) = ax 2 + bx + c, a < 0 d) Polynomial, not linear or quadratic
Oct 182:19 PM
6
171S4.1p Polynomial Functions and Models
313/??.
Unemployed.
The table below shows the number of unemployed in the United States from 1996 through 2006.
See TI tutorial at http://cfcc.edu/faculty/cmoore/TI83Modeling.htm
a) Use a graphing calculator to fit cubic and quartic functions to the data. Let x represent the number of years since 1996. b) Use the functions found in part (a) to estimate the number of unemployed in 2008. Compare the estimates and determine which model gives the more realistic estimate.
October 28, 2012
313/77.
Foreign Adoptions. The number of foreign adoptions in the United States has declined in recent years, as shown in the table below.
See TI tutorial at http://cfcc.edu/faculty/cmoore/TI83Modeling.htm
a) Use a graphing calculator to fit quadratic, cubic, and quartic functions to the data. Let x represent the number of years since 2000. Using R 2 values determine which function is the best fit.
b) Use the functions found in part (a) to estimate the number of U.S. foreign adoptions in 2010.
Oct 182:19 PM
313/78.
U.S. Farm Acreage. As the number of farms has decreased in the United States, the average size of the remaining farms has grown larger, as shown in the table below.
See TI tutorial at http://cfcc.edu/faculty/cmoore/TI83Modeling.htm
a) Use a graphing calculator to fit quadratic, cubic, and quartic functions to the data. Let x represent the number of years since 1900. Using R 2 values determine which function is the best fit.
b) Using the functions found in part (a) estimate the average acreage in 1955, in
1998, and in 2011.
Oct 182:19 PM
314/80. Dog Years. A dog’s life span is typically much shorter than that of a human. Age equivalents for dogs and humans are listed in the table below.
See TI tutorial at http://cfcc.edu/faculty/cmoore/TI83Modeling.htm
a) Use a graphing calculator to fit linear and cubic functions to the data. Which function has the better fit? [ Larger R 2 ] b) Using the function from part (a), estimate the equivalent human age for dogs that are 5, 10, and 15 years old.
Oct 182:19 PM
See TI tutorial at http://cfcc.edu/faculty/cmoore/TI83Modeling.htm
Oct 182:19 PM
7