1. Polynomials and Polynomial Functions
Chapter 6
Polynomials and Polynomial Functions

2. In this chapter, you will …
Learn to write and graph polynomial functions and to solve polynomial equations.
Learn to use important theorems about the number of solutions to polynomial equations.
Learn to solve problems involving permutations, combinations and binomial probability.

3. 6-1 Polynomial Functions
What you’ll learn …
To classify polynomials
To model data with polynomial functions

4. A monomial- “one term” is an expression that is a real number, a variable or a product of real numbers and variables. 13,    3x,    -57,   x²,   4y²,  -2xy,  or  520x²y²                     A binomial “two terms” is the sum of two monomials.  It has two unlike terms.  3x + 1,    x² - 4x,     2x + y,    or    y - y²   A trinomial “three Terms” is the sum of three monomials.  It has three unlike terms.  x2 + 2x + 1,     3x² - 4x + 10,      2x + 3y + 2   A polynomial “many terms” is the sum of one or more terms.  x2 + 2x,  3x3 + x² + 5x + 6, 4x - 6y + 8

5. The exponent or sum of exponents of the variable(s) in a term determines the degree of that term.
The terms in the polynomial must be in descending order by degree.
This order demonstrates the standard form of a polynomial. 2 3
2x -5x -2x + 5
Leading
Coefficient
Cubic
Term
Quadratic
Term
Linear
Term
Constant
Term

6. Name Using Number of Terms
Degree
Name Using Degree
Polynomial Example
Number of Terms
Name Using Number of Terms
Constant 6 1
monomial
Linear
x + 3 2
binomial
Quadratic
3x2 3
Cubic
2x3 -5x2 -2x
trinomial 4
Quartic
x4 + 3x2 5
Quintic
-2x5+3x2-x+4
polynomial

7. Example 1 Classifying Polynomials
Write each in standard form and classify it by degree and number of terms.
-7x + 5x4
x2 – 4x + 3x3 +2x
(4x)(6x + 5)

8. 6-2 Polynomials and Linear Factors
What you’ll learn …
To analyze the factored form of a polynomial
To write a polynomial function from its zeros

9. Cubic Function
A cubic polynomial is a polynomial of degree 3

10. Example 1 Writing a Polynomial in Standard Form
Write the expression (x+1)(x+2)(x+3) as a polynomial in standard form.
Write the expression (x+1)(x+1)(x+2) as a polynomial in standard form.

11. Example 2 Writing a Polynomial in Factored Form
Write 2x3 +10x2 + 12x in factored form.
Write 3x3 - 3x2 - 36x in factored form.

12. Example 4 Finding Zeros of a Polynomial Function
Find the zeros of
y= (x-2)(x+1)(x+3).
Then graph the function.
Find the zeros of
y= (x-7)(x-5)(x-3).
Then graph the function.

13. Example 5 Writing a Polynomial Function From Its Zeros
Write a polynomial function in standard form with zeros at -2, 3, and 3.
Write a polynomial function in standard form with zeros at -4, -2, and 1.

14. If a factor of a polynomial is repeated, then the zero is repeated
If a factor of a polynomial is repeated, then the zero is repeated. A repeated zero is called a multiple zero. A multiple zero has a multiplicity equal to the number of times the zero occurs.

15. Example 6 Finding the Multiplicity of a Zero
Find any multiple zeros of f(x)=x4 +6x3+8x2 and state the multiplicity.
Find any multiple zeros of f(x)=x3 - 4x2+4x and state the multiplicity.

16. Equivalent Statements about Polynomials
-4 is a solution of x2 +3x -4 =0.
-4 is an x-intercept of the graph of y= x2 +3x -4.
-4 is a zero of y= x2 +3x -4.
x+4 is a factor of x2 +3x -4.

17. 6-3 Dividing Polynomials
What you’ll learn …
To divide polynomials using long division
To divide polynomials using synthetic division

18. You can use polynomial division to help find all the factors (then, zeros) of a polynomial function. Division of polynomials is similar to numerical long division.
2 56
x 2x2

19. Example 1a Polynomial Long Division
Divide x2 +3x – 12 by x - 3

20. Example 1a Polynomial Long Division
Divide x3 +3x2 +7x – 12 by x - 3

21. Example 1a Polynomial Long Division
Divide 6x3 +3x – 12 by 3x - 6

22. Example 2 Checking Factors
Determine whether x+4 is a factor of each polynomial
x2 + 6x + 8
x3 + 3x2 -6x - 7
To check (divisor) (quotient) + r = dividend

23. Synthetic division is a simplified process in which you omit all variables and exponents. By reversing the sign of the divisor, you can add throughout the process instead of subtracting.

24. Example 3a Using Synthetic Division
Use synthetic division to divide x3 – 4x2 +2x – 1 by x +1

25. Example 3b Using Synthetic Division
Use synthetic division to divide x3 + 4x2 + x – 6 by x +1

26. Check for Understanding
Use synthetic division to divide x3 - 2x2 - 5x + 6 by x + 2

27. Remainder Theorem
If a polynomial P(x) of degree n>1 is divided by (x-a). Where a is a constant, then the remainder is P(a).

28. Example 5a Evaluating a Polynomial by Synthetic Division
Use synthetic division to find P(-4) for P(x) = x4 - 5x2 + 4x + 12

29. Example 5b Evaluating a Polynomial by Synthetic Division
Use synthetic division to find P(-1) for P(x) = 2x4 + 6x3 – x2 - 60

30. 6-4 Solving Polynomial Equations
What you’ll learn …
To solve polynomials equations by graphing
To solve polynomials equations by factoring

31. Example 1 Solving by Graphing
Solve x3 + 3x2 = x + 3
Solve x3 - 19x = -2x2 + 20

32. Sometimes you can solve polynomial equations by factoring the polynomial and using the factor Theorem. Recall that a quadratic expression that is the difference of squares has a special factoring pattern. Similarly, a cubic expression may be the sum of cubes or the difference of cubes.
Sum and Differences of Cubes
a3 + b3 = (a + b)(a2 – ab + b2)
a3 - b3 = (a - b)(a2 + ab + b2)

33. Example 3 Factoring a Sum of Cubes
Factor 8x3 + 1
Factor 64x3 + 27

34. Example 3 Factoring a Difference of Cubes
Factor 8x3 - 27
Factor 125x3 - 64

35. Example 4 Solving a Polynomial Equation
Factor x3 + 8 = 0
Factor 27x3 + 1

36. Example 5 Factoring by Using a Quadratic Form
Factor x4 - 2x2 – 8 = 0
Factor x4 + 7x2 + 6 = 0

37. 6-5 Theorems About Roots of Polynomial Equations
What you’ll learn …
To solve equations using the Rational Root Theorem
To use the Irrational Root Theorem and the Imaginary Root Theorem
1.02 Define and compute with complex numbers.
1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

38. Consider the equivalent equations….
x3 – 5x2 -2x +24 = 0 and (x+2)(x-3)(x-4) =0
-2, 3 and 4 are the roots of the equation.
The product of -2,3 and 4 is 24.
Notice that all the roots are factors of the constant term 24.
In general, if the coefficients in a polynomial equation are integers, then any integer root of the equation is a factor of the constant term.

39. Both the constant and the leading coefficient of a polynomial can play a key role in identifying the rational roots of the related polynomial equation. The role is expressed in the Rational Root Theorem. p q
If is in simplest form and is a rational root of the polynomial equation with integer coefficients, then p must be a factor of the constant term and q must be a factor of the leading coefficient.

40. Example 1a Finding Rational Roots
Steps
List the possible rational roots of the leading coefficient and the constant.
Test each possible root.
x3 - 4x2 - 2x + 8 = 0

41. Example 1b Finding Rational Roots
Steps
List the possible rational roots of the leading coefficient and the constant.
Test each possible root.
2x3 - x2 +2x - 1 = 0

42. Example 1c Finding Rational Roots
Steps
List the possible rational roots of the leading coefficient and the constant.
Test each possible root.
x3 - 2x2 - 5x + 10 = 0

43. Example 1d Finding Rational Roots
Steps
List the possible rational roots of the leading coefficient and the constant.
Test each possible root.
3x3 + x2 - x + 1 = 0

44. In Chapter 5 you learned to find irrational solutions to quadratic equations. For example, by the Quadratic Formula, the solutions of x2 – 4x -1 =0 are 2+√5 and 2 - √5.
Number pairs of the form a+√b and a-√b are called conjugates.

45. You can often use conjugates to find the irrational roots of a polynomial equation.
Irrational Root Theorem
Let a and b be rational numbers and let √b be an irrational number. If a+ √b is a root of a polynomial equation with rational coefficients, then the conjugate a- √b also is a root.

46. Example 3 Finding Irrational Roots
A polynomial equation with integer coefficients has
the following roots. Find two additional roots..
1 + √3 and -√11
2 - √7 and √5

47. Imaginary Root Theorem
Number pairs of the form a+bi and a-bi are complex conjugates. You can use complex conjugates to find an equation’s imaginary roots.
Imaginary Root Theorem
If the imaginary number a+bi is a root of a polynomial equation with real coefficients then the conjugate a-bi also is a root.

48. Example 4 Finding Imaginary Roots
A polynomial equation with integer coefficients has
the following roots. Find two additional roots..
3i and -2 + i
3 - i and 2i

49. Example 5a Writing a Polynomial Equation from its Roots
Find a third degree polynomial equation with
rational coefficients that has roots -1 and 2-i.
Steps
Find the other root using the Imaginary Root Theorem.
Write the factored form of the polynomial using the Factor Theorem.
Multiply the factors.

50. Example 5b Writing a Polynomial Equation from its Roots
Find a third degree polynomial equation with
rational coefficients that has roots 3 and 1+i.
Steps
Find the other root using the Imaginary Root Theorem.
Write the factored form of the polynomial using the Factor Theorem.
Multiply the factors.

51. Example 5c Writing a Polynomial Equation from its Roots
Find a fourth degree polynomial equation with
rational coefficients that has roots i and 2i.
Steps
Find the other root using the Imaginary Root Theorem.
Write the factored form of the polynomial using the Factor Theorem.
Multiply the factors.

52. 6-6 The Fundamental Theorem of Algebra
What you’ll learn …
To use the Fundamental Theorem of Algebra in solving polynomial equations with complex roots
1.02 Define and compute with complex numbers.
1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

53. You have solved polynomial equations and found that their roots are included in the set of complex numbers. That is, the roots have been integers, rational numbers, irrational numbers and imaginary numbers.
But can all polynomial equations be solved using complex numbers?

54. In 1799, the German mathematician Carl Friedrich Gauss proved that the answer to this question is yes. The roots of every polynomial equation, even those with imaginary coefficients, are complex numbers.
The answer is so important that his theorem is called the Fundamental Theorem of Algebra.
Carl Friedrich Gauss

55. Fundamental Theorem of Algebra
If P(x) is a polynomial of degree n>1 with complex coefficients, then P(x) = 0 has at least one complex root.
Corollary
Including imaginary roots and multiple roots, an nth degree polynomial equation has exactly n roots; the related polynomial function has exactly n zeros.

56. Example 1a Using the Fundamental Theorem of Algebra
Find the number of complex roots, the possible number of real roots and possible number of rational roots.
x4 - 3x3 + x2 – x +3 = 0

57. Example 1b Using the Fundamental Theorem of Algebra
Find the number of complex roots, the possible number of real roots and possible number of rational roots.
x3 - 2x2 + 4x -8 = 0

58. Example 1c Using the Fundamental Theorem of Algebra
Find the number of complex roots, the possible number of real roots and possible number of rational roots.
x5 + 3x4 - x - 3 = 0

59. 6-8 The Binomial Theorem
What you’ll learn …
To use Pascal’s Triangle
To use the Binomial Theorem
1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

60. You have learned to multiply binomials using the FOIL method and the Distributive Property. If you are raising a single binomial to a power, you have another option for finding the product.
Consider the expansion of several binomials. To expand a binomial being raised to a power, first multiply; then write the result as a polynomial in standard form.

61. Pascal’s Triangle

62. (a + b)3 = (a + b) (a + b) (a + b)
a2 + 2ab + b2
The coefficients of the product are 1,2 1.
(a + b)3 = (a + b) (a + b) (a + b)
a3 + 3a2b + 3ab2 + b3
The coefficients of the product are 1,3,3,1.

63. Pascal’s Triangle

64. Example 1a Using Pascal’s Triangle
Expand (a+b)8

65. Example 1b Using Pascal’s Triangle
Expand (x - 2)4

66. Example 1c Using Pascal’s Triangle
Expand (m + 3)5

67. Example 1d Using Pascal’s Triangle
Expand (3 – 2x)6

68. Quadratic Inequalities

69. Quadratics Before we get started let’s review.
A quadratic equation is an equation that can
be written in the form ,
where a, b and c are real numbers and a cannot equal
zero.
In this lesson we are going to discuss quadratic
inequalities.

70. Quadratic Inequalities
What do they look like?
Here are some examples:

71. Quadratic Inequalities
When solving inequalities we are trying to
find all possible values of the variable
which will make the inequality true.
Consider the inequality
We are trying to find all the values of x for which the
quadratic is greater than zero or positive.

72. Solving a quadratic inequality
We can find the values where the quadratic equals zero
by solving the equation,

73. Solving a quadratic inequality
For the quadratic inequality,
we found zeros 3 and –2 by solving the equation
. Put these values on a number line and we can see three intervals that we will test in the inequality. We will test one value from each interval. -2 3

74. Solving a quadratic inequality
Interval
Test Point
Evaluate in the inequality
True/False

75. Example 2:
Solve
First find the zeros by solving the equation,

76. Example 2:
Now consider the intervals around the zeros and test a value from each interval in the inequality.
The intervals can be seen by putting the zeros on a number line.
1/2 1

77. Example 2:
Interval
Test Point
Evaluate in Inequality
True/False

78. Summary In general, when solving quadratic inequalities
Find the zeros by solving the equation you get when you replace the inequality symbol with an equals.
Find the intervals around the zeros using a number line and test a value from each interval in the number line.
The solution is the interval or intervals which make the inequality true.

83. In this chapter, you should have …
Learned to write and graph polynomial functions and to solve polynomial equations.
Learned to use important theorems about the number of solutions to polynomial equations.
Learned to solve problems involving permutations, combinations and binomial probability.