1. Polynomial and Rational
College Algebra
Chapter 4
Polynomial and Rational
Functions

2. 4.1 Polynomial Long Division and Synthetic Division
When the division cannot be completed by factoring, polynomial long division is used and closely resembles whole number division
In the division process, zero “place holders” are sometimes used to ensure that like place values will “line up” as we carry out the algorithm

3. 4.1 Polynomial Long Division and Synthetic Division

4. 4.1 Polynomial Long Division and Synthetic Division
How to handle a remainder

5. 4.1 Polynomial Long Division and Synthetic Division

6. The same idea holds for polynomials
4.1 Polynomial Long Division and Synthetic Division
If one number divides evenly into another, it must be a factor of the original number
The same idea holds for polynomials
This means that division can be used as a tool for factoring
We need to do two things first
Find a more efficient method for division
Find divisors that give a remainder of zero

7. 5 1 -2 -13 -17 Synthetic Division 5 15 10 2 -7 1 3 remainder
4.1 Polynomial Long Division and Synthetic Division
Synthetic Division 5 5 15 10 2 -7 1 3
remainder
Multiply in the diagonal direction, add in the vertical direction
Explanation of why it works is on pg 376

8. 4.1 Polynomial Long Division and Synthetic Division

9. 4.1 Polynomial Long Division and Synthetic Division

10. Principal of Factorable Polynomials
4.1 Polynomial Long Division and Synthetic Division
Synthetic Division and Factorable Polynomials
Principal of Factorable Polynomials
Given a polynomial of degree n>1 with integer coefficients and a lead coefficient of 1 or -1, the linear factors of the polynomial must be of the form (x-p) where p is a factor of the constant term.
Use synthetic division to help factor
Hint: Start with what is easiest 1 1
-10 -4 24

11. Synthetic Division and Factorable Polynomials
4.1 Polynomial Long Division and Synthetic Division
Synthetic Division and Factorable Polynomials

12. What values of k will make x-3 a factor of
4.1 Polynomial Long Division and Synthetic Division
What values of k will make x-3 a factor of

13. 4.1 Polynomial Long Division and Synthetic Division
Homework pg

14. Use the remainder theorem to find the value of H(-5) for
4.2 The Remainder and Factor Theorems
The Remainder Theorem
If a polynomial P(x) is divided by a linear factor (x-r), the remainder is identical to P(r) – the original function evaluated at r.
Use the remainder theorem to find the value of H(-5) for

15. Use the remainder theorem to find the value of P(1/2) for
4.2 The Remainder and Factor Theorems
Use the remainder theorem to find the value of P(1/2) for

16. The Factor Theorem Given P(x) is a polynomial,
4.2 The Remainder and Factor Theorems
The Factor Theorem
Given P(x) is a polynomial,
If P(r) = 0, then (x-r) is a factor of P(x).
If (x-r) is a factor of P(x), then P(r) = 0
Use the factor theorem to find a cubic polynomial with these three roots:

17. 4.2 The Remainder and Factor Theorems
A polynomial P with integer coefficients has the zeros and degree indicated. Use the factor theorem to write the function in factored and standard form.

18. 4.2 The Remainder and Factor Theorems
Complex numbers, coefficients, and the Remainder and Factor Theorems
Show x=2i is a zero of:

19. Complex Conjugates Theorem
4.2 The Remainder and Factor Theorems
Complex Conjugates Theorem
Given polynomial P(x) with real number coefficients, complex solutions will occur in conjugate pairs.
If a+bi, b≠0, is a solution, then a-bi must also be a solution.

20. Polynomial zeroes theorem
4.2 The Remainder and Factor Theorems
Roots of multiplicity
Some equations produce repeated roots.
Polynomial zeroes theorem
A polynomial equation of degree n has exactly n roots, (real and complex) where roots of multiplicity m are counted m times.

21. 4.2 The Remainder and Factor Theorems
Homework pg

22. The Fundamental Theorem of Algebra
4.3 The Zeroes of Polynomial Functions
The Fundamental Theorem of Algebra
Every complex polynomial of degree n≥1 has at least one complex root.
Our search for a solution will not be fruitless or wasted, solutions for all polynomials exist.
The fundamental theorem combined with the factor theorem enables to state the linear factorization theorem.

23. Linear factorization theorem
4.3 The Zeroes of Polynomial Functions
Linear factorization theorem
Every complex polynomial of degree n ≥ 1 can be written as the product of a nonzero constant and exactly n linear factors
THE IMPACT
Every polynomial equation, real or complex, has exactly n roots, counting roots of multiplicity

24. 4.3 The Zeroes of Polynomial Functions
Find all zeroes of the complex polynomial C, given x = 1-I is a zero. Then write C in completely factored form:

25. The Intermediate Value Theorem (IVT)
4.3 The Zeroes of Polynomial Functions
The Intermediate Value Theorem (IVT)
Given f is a polynomial with real coefficients, if f(a) and f(b) have opposite signs, there is at least one value r between a and b such that f(r)=0
HOW DOES THIS HELP???
Finding factors of polynomials

26. The Rational Roots Theorem (RRT)
4.3 The Zeroes of Polynomial Functions
The Rational Roots Theorem (RRT)
Given a real polynomial P(x) with degree n ≥ 1 and integer coefficients, the rational roots of P (if they exist) must be of the form p/q, where p is a factor of the constant term and q is a factor of the lead coefficient (p/q must be written in lowest terms)
List the possible rational roots for

27. 4.3 The Zeroes of Polynomial Functions
Tests for 1 and -1
If the sum of all coefficients is zero, x = 1 is a rood and (x-1) is a factor.
After changing the sign of all terms with odd degree, if the sum of the coefficients is zero, then x = -1 is a root and (x+1) is a factor.

28. 4.3 The Zeroes of Polynomial Functions
Homework pg

29. THE END BEHAVIOR OF A POLYNOMIAL GRAPH
4.4 Graphing Polynomial Functions
THE END BEHAVIOR OF A POLYNOMIAL GRAPH
If the degree of the polynomial is odd, the ends will point in opposite directions:
1. Positive lead coefficient: down on left, up on right (like y=x3)
Negative lead coefficient: up on left, down on right (like y=-x3)
If the degree of the polynomial is even, the ends will point in the same direction:
Positive lead coefficient: up on left, up on right (like y=x2)
Negative lead coefficient: down on left, down on right (like y=-x2)

30. Attributes of polynomial graphs with roots of multiplicity
4.4 Graphing Polynomial Functions
Attributes of polynomial graphs with roots of multiplicity
Zeroes of odd multiplicity will “cross through” the x-axis
Zeroes of even multiplicity will “bounce” off the x-axis
Cross through
bounce

31. Estimate the equation based on the graph
4.4 Graphing Polynomial Functions
Estimate the equation based on the graph

32. g(x) = (x - 2)² (x + 1)³ (x - 1)²
4.4 Graphing Polynomial Functions
Estimate the equation based on the graph
g(x) = (x - 2)² (x + 1)³ (x - 1)²

33. Guidelines for Graphing Polynomial Functions
Determine the end behavior of the graph
Find the y-intercept f(0) = ?
Find the x-intercepts using any combination of the rational root theorem, factor and remainder theorems, factoring, and the quadratic formula.
Use the y-intercepts, end behavior, the multiplicity of each zero, and a few mid-interval points to sketch a smooth, continuous curve.

34. Sketch the graph of Down, Down F(0) = -12
4.4 Graphing Polynomial Functions
Sketch the graph of
Determine the end behavior of the graph
Find the y-intercept f(0) = ?
Find the x-intercepts using any combination of the rational root theorem, factor and remainder theorems, factoring, and the quadratic formula.
Use the y-intercepts, end behavior, the multiplicity of each zero, and a few mid-interval points to sketch a smooth, continuous curve.
Down, Down
F(0) = -12
bounce
Cut through
Cut through

35. f(x) = x⁶ - 2 x⁵ - 4 x⁴ + 8 x³ 4.4 Graphing Polynomial Functions
Determine the end behavior of the graph
Find the y-intercept f(0) = ?
Find the x-intercepts using any combination of the rational root theorem, factor and remainder theorems, factoring, and the quadratic formula.
Use the y-intercepts, end behavior, the multiplicity of each zero, and a few mid-interval points to sketch a smooth, continuous curve.

36. 4.4 Graphing Polynomial Functions
Homework pg

37. Vertical Asymptotes of a Rational Function
4.5 Graphing Rational Functions
Vertical Asymptotes of a Rational Function
Given is a rational function in lowest
terms, vertical asymptotes will occur at the real zeroes of g
The “cross” and “bounce” concepts used for polynomial graphs can also be applied to rational graphs
cross
bounce

38. x- and y-intercepts of a rational function
4.5 Graphing Rational Functions
x- and y-intercepts of a rational function
Given is in lowest terms, and x = 0 in the domain of r,
To find the y-intercept, substitute 0 for x and simplify. If 0 is not in the domain, the function has no y-intercept
To find the x-intercept(s), substitute 0 for f(x) and solve. If the equation has no real zeroes, there are no x-intercepts.
Determine the x- and y-intercepts for the function

39. Determine the x- and y-intercepts for the function
4.5 Graphing Rational Functions
Determine the x- and y-intercepts for the function
Y-intercept
No x-intercept

40. Given is a rational function in lowest
4.5 Graphing Rational Functions
Given is a rational function in lowest
terms, where the lead term of f is axn and the lead term of g is bxm
Polynomial f has degree n, polynomial g has degree m
If n<m, the graph of h has a horizontal asymptote at y=0 (the x-axis)
If n=m, the graph of h has a horizontal asymptote at y=a/b (the ratio of lead coefficients)
If n>m, the graph of h has no horizontal asymptote

41. Guidelines for graphing rational functions pg 428
Find the y-intercept [evaluate r(0)]
Locate vertical asymptotes x=h [solve g(x) = 0]
Find the x-intercepts (if any) [solve f(x) = 0]
Locate the horizontal asymptote y = k (check degree of numerator and denominator)
Determine if the graph will cross the horizontal asymptote [solve r(x) = k from step 4
If needed, compute the value of any “mid-interval” points needed to round-out the graph
Draw the asymptotes, plot the intercepts and additional points, and use intervals where r(x) changes sign to complete the graph
Given is a rational function in lowest
terms, where the lead term of f is axn and the lead term of g is bxm

42. 4.5 Graphing Rational Functions
Find the y-intercept [evaluate r(0)]
Locate vertical asymptotes x=h [solve g(x) = 0]
Find the x-intercepts (if any) [solve f(x) = 0]
Locate the horizontal asymptote y = k (check degree of numerator and denominator)
Determine if the graph will cross the horizontal asymptote [solve r(x) = k from step 4
If needed, compute the value of any “mid-interval” points needed to round-out the graph
Draw the asymptotes, plot the intercepts and additional points, and use intervals where r(x) changes sign to complete the graph

43. 4.5 Graphing Rational Functions
Homework pg

44. Given is a rational function in lowest
4.5 Graphing Rational Functions
Given is a rational function in lowest
terms, where the lead term of f is axn and the lead term of g is bxm
Polynomial f has degree n, polynomial g has degree m
If n<m, the graph of h has a horizontal asymptote at y=0 (the x-axis)
If n=m, the graph of h has a horizontal asymptote at y=a/b (the ratio of lead coefficients)
If n>m, the graph of h has no horizontal asymptote

45. Oblique and nonlinear asymptotes
4.6 Additional Insights into Rational Functions
Oblique and nonlinear asymptotes
Given is a rational function in lowest
terms, where the degree of f is greater than the degree of g. The graph will have an oblique or nonlinear asymptote as determined by q(x), where q(x) is the quotient of

46. 4.6 Additional Insights into Rational Functions

47. Choose one application problem
4.6 Additional Insights into Rational Functions
Choose one application problem

48. 4.6 Additional Insights into Rational Functions
Homework pg

49. Solving Polynomial Inequalities
4.7 Polynomial and Rational Inequalities – An Analytical View
Solving Polynomial Inequalities
Given f(x) is a polynomial in standard form pg 452
Use any combination of factoring, tests for 1 and -1, the RRT and synthetic division to write P in factored form, noting the multiplicity of each zero.
Plot the zeroes on a number line (x-axis) and determine if the graph crosses (odd multiplicity) or bounces (even multiplicity) at each zero. Recall that complex zeroes from irreducible quadratic factors can be ignored.
Use end behavior, the y-intercept, or a test point to determine the sign of the function in a given interval, then label all other intervals as P(x) < 0 or P(x) > 0 by analyzing the multiplicity of neighboring zeroes.
State the solution using interval notation, noting strict/non-strict inequalities.

50. End behavior is down/up
4.7 Polynomial and Rational Inequalities – An Analytical View
Use any combination of factoring, tests for 1 and -1, the RRT and synthetic division to write P in factored form, noting the multiplicity of each zero.
Plot the zeroes on a number line (x-axis) and determine if the graph crosses (odd multiplicity) or bounces (even multiplicity) at each zero. Recall that complex zeroes from irreducible quadratic factors can be ignored.
Use end behavior, the y-intercept, or a test point to determine the sign of the function in a given interval, then label all other intervals as P(x) < 0 or P(x) > 0 by analyzing the multiplicity of neighboring zeroes.
State the solution using interval notation, noting strict/non-strict inequalities.
Synthetic division
cross
bounce
End behavior is down/up up
down
f(x) > 0
f(x) < 0
f(x) < 0

51. Add coefficients 1+1+-5+3=0
4.7 Polynomial and Rational Inequalities – An Analytical View
Use any combination of factoring, tests for 1 and -1, the RRT and synthetic division to write P in factored form, noting the multiplicity of each zero.
Plot the zeroes on a number line (x-axis) and determine if the graph crosses (odd multiplicity) or bounces (even multiplicity) at each zero. Recall that complex zeroes from irreducible quadratic factors can be ignored.
Use end behavior, the y-intercept, or a test point to determine the sign of the function in a given interval, then label all other intervals as P(x) < 0 or P(x) > 0 by analyzing the multiplicity of neighboring zeroes.
State the solution using interval notation, noting strict/non-strict inequalities.
Test for 1 and -1
Add coefficients =0
Means that x=1 is a root
End behavior down/up
bounce
cross
f(x) < 0
f(x) > 0
f(x) > 0

52. The graph will change signs at x = 2, -3, and 7/4
4.7 Polynomial and Rational Inequalities – An Analytical View
The graph will change signs at x = 2, -3, and 7/4
The y-intercept is 7/6 which is positive
above
above
below
below

53. 4.7 Polynomial and Rational Inequalities – An Analytical View
above
above
below

54. 4.7 Polynomial and Rational Inequalities – An Analytical View
Homework pg

55. Chapter 4 Review

56. Chapter 4 Review

57. Chapter 4 Review
Use synthetic division to show that (x+7) is a factor of 2x4+13x3-6x2+9x+14

58. Factor and state roots of multiplicity
Chapter 4 Review
Factor and state roots of multiplicity

59. State an equation for the given graph
Chapter 4 Review
State an equation for the given graph

60. State an equation for the given graph
Chapter 4 Review
State an equation for the given graph

61. Chapter 4 Review
Graph

62. Divide using long division
Chapter 4 Review
Trashketball Review
Divide using long division

63. Use synthetic division to divide
Chapter 4 Review
Trashketball Review
Use synthetic division to divide

64. Show the indicated value is a zero of the function
Chapter 4 Review
Trashketball Review
Show the indicated value is a zero of the function

65. Show the indicated value is a zero of the function
Chapter 4 Review
Trashketball Review
Show the indicated value is a zero of the function

66. Find all the zeros of the function
Chapter 4 Review
Trashketball Review
Find all the zeros of the function
Real root x=3
Complex roots x=±2i

67. Find all the zeros of the function
Chapter 4 Review
Trashketball Review
Find all the zeros of the function

68. Chapter 4 Review
Trashketball Review
State end behavior, y-intercept, and list the possible rational roots for each function

69. Chapter 4 Review
Trashketball Review
State end behavior, y-intercept, and list the possible rational roots for each function

70. Chapter 4 Review
Trashketball Review
Sketch the Graph using the degree, end behavior, x- y-intercept, zeroes of multiplicity and midinterval points

71. Chapter 4 Review
Trashketball Review
Sketch the Graph using the degree, end behavior, x- y-intercept, zeroes of multiplicity and midinterval points

72. Graph using guidelines for graphing rational functions
Chapter 4 Review
Trashketball Review
Graph using guidelines for graphing rational functions

73. Graph using guidelines for graphing rational functions
Chapter 4 Review
Trashketball Review
Graph using guidelines for graphing rational functions