1. 4.1 Objective:
Students will look at polynomial functions of degree greater than 2, approximate the zeros, and interpret graphs.

2. Functions of degree greater than 2
4.1 Polynomial
Functions of degree greater than 2

3. A polynomial function is a function of the form:
Remember integers are … –2, -1, 0, 1, 2 … (no decimals or fractions) so positive integers would be 0, 1, 2 …
A polynomial function is a function of the form:
n must be a positive integer
All of these coefficients are real numbers
The degree of the polynomial is the largest power on any x term in the polynomial.

4. Determine which of the following are polynomial functions
Determine which of the following are polynomial functions. If the function is a polynomial, state its degree.
A polynomial of degree 4.
We can write in an x0 since this = 1.
x 0
A polynomial of degree 0.
Not a polynomial because of the square root since the power is NOT an integer
Not a polynomial because of the x in the denominator since the power is negative

5. Graphs of polynomials are smooth and continuous.
No gaps or holes, can be drawn without lifting pencil from paper
No sharp corners or cusps
This IS the graph of a polynomial
This IS NOT the graph of a polynomial

6. Let’s look at the graph of where n is an even integer.
and grows steeper on either side
Notice each graph looks similar to x2 but is wider and flatter near the origin between –1 and 1
The higher the power, the flatter and steeper

7. Let’s look at the graph of where n is an odd integer.
Notice each graph looks similar to x3 but is wider and flatter near the origin between –1 and 1
and grows steeper on either side
The higher the power, the flatter and steeper

8. Let’s graph Translates up 2 Reflects about the x-axis
Looks like x2 but wider near origin and steeper after 1 and -1
So as long as the function is a transformation of xn, we can graph it, but what if it’s not? We’ll learn some techniques to help us determine what the graph looks like in the next slides.

9. HAND BEHAVIOUR OF A GRAPH
RIGHT
LEFT
HAND BEHAVIOUR OF A GRAPH
The degree of the polynomial along with the sign of the coefficient of the term with the highest power will tell us about the left and right hand behaviour of a graph.

10. Even degree polynomials rise on both the left and right hand sides of the graph (like x2) if the coefficient is positive. The additional terms may cause the graph to have some turns near the center but will always have the same left and right hand behaviour determined by the highest powered term.
left hand behaviour: rises
right hand behaviour: rises

11. Even degree polynomials fall on both the left and right hand sides of the graph (like - x2) if the coefficient is negative.
turning points in the middle
left hand behaviour: falls
right hand behaviour: falls

12. Odd degree polynomials fall on the left and rise on the right hand sides of the graph (like x3) if the coefficient is positive.
turning Points in the middle
right hand behaviour: rises
left hand behaviour: falls

13. Odd degree polynomials rise on the left and fall on the right hand sides of the graph (like x3) if the coefficient is negative.
turning points in the middle
left hand behaviour: rises
right hand behaviour: falls

14. How do we determine what it looks like near the middle?
A polynomial of degree n can have at most n-1 turning points (so whatever the degree is, subtract 1 to get the most times the graph could turn).
doesn’t mean it has that many turning points but that’s the most it can have
Let’s determine left and right hand behaviour for the graph of the function:
degree is 4 which is even and the coefficient is positive so the graph will look like x2 looks off to the left and off to the right.
How do we determine what it looks like near the middle?
The graph can have at most 3 turning points

15. To find the x intercept we put 0 in for y.
x and y intercepts would be useful and we know how to find those. To find the y intercept we put 0 in for x.
To find the x intercept we put 0 in for y.
Finally we need a smooth curve through the intercepts that has the correct left and right hand behavior. To pass through these points, it will have 3 turns (one less than the degree so that’s okay)
(0,30)

16. We found the x intercept by putting 0 in for f(x) or y (they are the same thing remember). So we call the x intercepts the zeros of the polynomial since it is where it = 0. These are also called the roots of the polynomial.

17. 4.2 Dividing Polynomials
© 2002 by Shawna Haider

18. 4.2 Properties of Division
Objective: We will learn how to use synthetic division which will help us in finding imaginary roots of polynomial functions.

19. 3x3 4x2 x 2 1 1 1 1 Dividing by a Monomial
If the divisor only has one term, split the polynomial up into a fraction for each term.
divisor
Now reduce each fraction.
3x3
4x2 x 2 1 1 1 1

20. Subtract (which changes the sign of each term in the polynomial)
Long Division
If the divisor has more than one term, perform long division. You do the same steps with polynomial division as with integers. Let's do two problems, one with integers you know how to do and one with polynomials and copy the steps.
Subtract (which changes the sign of each term in the polynomial)
Now multiply by the divisor and put the answer below. x
+ 11 2 1
Bring down the next number or term
Multiply and put below
Now divide 3 into or x into 11x
First divide 3 into or x into x2
x x2 + 8x - 5
Remainder added here over divisor 64
x2 – 3x
subtract 5 8
11x
- 5 32
11x - 33
This is the remainder 26 28
So we found the answer to the problem x2 + 8x – 5  x – 3 or the problem written another way:

21. Remainder added here over divisor
Let's Try Another One
If any powers of terms are missing you should write them in with zeros in front to keep all of your columns straight.
Subtract (which changes the sign of each term in the polynomial) y
- 2
Write out with long division including 0y for missing term
Multiply and put below
Bring down the next term
Multiply and put below
y y2 + 0y + 8
Divide y into y2
Divide y into -2y
Remainder added here over divisor
y2 + 2y
subtract
-2y
+ 8
- 2y - 4
This is the remainder 12

22. - 3 1 6 8 -2 - 3 - 9 3 1 x2 + x 3 - 1 1 1 Synthetic Division
There is a shortcut for long division as long as the divisor is x – k where k is some number. (Can't have any powers on x).
Set divisor = 0 and solve. Put answer here.
x + 3 = 0 so x = - 3 1
- 3
Multiply these and put answer above line in next column
Multiply these and put answer above line in next column
Multiply these and put answer above line in next column
Bring first number down below line
- 3
Add these up
- 9 3
Add these up
Add these up 1
x2 + x 3
- 1 1
This is the remainder
Put variables back in (one x was divided out in process so first number is one less power than original problem).
So the answer is:
List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put in a 0.

23. Let's try another Synthetic Division
0 x3
0 x
Set divisor = 0 and solve. Put answer here.
x - 4 = 0 so x = 4 1 4
Multiply these and put answer above line in next column
Multiply these and put answer above line in next column
Multiply these and put answer above line in next column
Multiply these and put answer above line in next column
Bring first number down below line 4
Add these up 16 48
192
Add these up
Add these up
Add these up 1
x3 + x x + 4 12 48
198
This is the remainder
Now put variables back in (remember one x was divided out in process so first number is one less power than original problem so x3).
So the answer is:
List all coefficients (numbers in front of x's) and the constant along the top. Don't forget the 0's for missing terms.

24. Let's try a problem where we factor the polynomial completely given one of its factors.
You want to divide the factor into the polynomial so set divisor = 0 and solve for first number.
- 2
Multiply these and put answer above line in next column
Multiply these and put answer above line in next column
Multiply these and put answer above line in next column
Bring first number down below line
- 8
Add these up 50
Add these up
Add these up
No remainder so x + 2 IS a factor because it divided in evenly 4
x2 + x
- 25
Put variables back in (one x was divided out in process so first number is one less power than original problem).
So the answer is the divisor times the quotient:
List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put in a 0.
You could check this by multiplying them out and getting original polynomial

25. 4.3 Zeros of Polynomials Objective:
Students will find the zeros of a polynomial and be able to discuss how many zeros are real & imaginary, positive & negative, rational & irrational and large or small in value.

26. 4.3 Zeros of Polynomials Can you find the zeros of the polynomial?
There are repeated factors. (x-1) is to the 3rd power so it is repeated 3 times. If we set this equal to zero and solve we get 1. We then say that 1 is a zero of multiplicity 3 (since it showed up as a factor 3 times).
What are the other zeros and their multiplicities?
-2 is a zero of multiplicity 2
3 is a zero of multiplicity 1

27. What would the left and right hand behavior be?
So knowing the zeros of a polynomial we can plot them on the graph. If we know the multiplicity of the zero, it tells us whether the graph crosses the x axis at this point (odd multiplicities CROSS) or whether it just touches the axis and turns and heads back the other way (even multiplicities TOUCH). Let’s try to graph:
What would the left and right hand behavior be?
You don’t need to multiply this out but figure out what the highest power on an x would be if multiplied out. In this case it would be an x3. Notice the negative out in front.
What would the y intercept be?
1 of mult. 1 (so crosses axis at 1)
-2 of mult. 2 (so touches at 2)
(0, 4)
Find the zeros and their multiplicity

28. Steps for Graphing a Polynomial
Determine left and right hand behaviour by looking at the highest power on x and the sign of that term.
Determine maximum number of turning points in graph by subtracting 1 from the degree.
Find and plot y intercept by putting 0 in for x
Find the zeros (x intercepts) by setting polynomial = 0 and solving.
Determine multiplicity of zeros.
Join the points together in a smooth curve touching or crossing zeros depending on multiplicity and using left and right hand behavior as a guide.

29. Find and plot y intercept by putting 0 in for x
Let’s graph:
Determine left and right hand behavior by looking at the highest power on x and the sign of that term.
Find and plot y intercept by putting 0 in for x
Determine maximum number of turns in graph by subtracting 1 from the degree.
Find the zeros (x intercepts) by setting polynomial = 0 and solving.
Join the points together in a smooth curve touching or crossing zeros depending on multiplicity and using left and right hand behaviour as a guide.
Determine multiplicity of zeros.
0 multiplicity 2 (touches) 3 multiplicity 1 (crosses) -4 multiplicity 1 (crosses)
Zeros are: 0, 3, -4
Multiplying out, highest power would be x4
Degree is 4 so maximum number of turns is 3
Here is the actual graph. We did pretty good. If we’d wanted to be more accurate on how low to go before turning we could have plugged in an x value somewhere between the zeros and found the y value. We are not going to be picky about this though since there is a great method in calculus for finding these maxima and minima.

30. What is we thought backwards
What is we thought backwards? Given the zeros and the degree can you come up with a polynomial? Find a polynomial of degree 3 that has zeros –1, 2 and 3.
What would the function look like in factored form to have the zeros given above?
Multiply this out to get the polynomial. FOIL two of them and then multiply by the third one.

31. 4.4 Complex and Rational Zeros of Polynomials
Objective:
In this lesson, we will:
Learn that imaginary roots always come in pairs and those pairs are conjugates
Find the possible and actual rational zeros of a polynomial function

32. 4.4 Complex and Rational Zeros of Polynomials
Using the Rational Root Theorem to Predict the Rational Roots of a Polynomial

33. Find the Roots of a Polynomial
For higher degree polynomials, finding the complex roots (real and imaginary) is easier if we know one of the roots.
The table below can help get you started. Complete the table below:

34. The Rational Root Theorem
The Rational Root Theorem gives us a tool to predict
the Values of Rational Roots:

35. List the Possible Rational Roots
For the polynomial:
All possible values of:
All possible Rational Roots of the form p/q:

36. Narrow the List of Possible Roots
For the polynomial:
Graph the polynomial:
# + real zeros: 1
# - real zeros: 0
# imag. zeros: 2
All possible Rational Roots of the form p/q:

37. Find a Root That Works For the polynomial:
Substitute each of our possible rational roots into f(x). If a value, a, is a root, then f(a) = 0. (Roots are solutions to an equation set equal to zero!)

38. Find the Other Roots
Now that we know one root is x = 3, do the other two roots have to be imaginary? What other category have we left out?
To find the other roots, divide the factor that we know into the original polynomial:

39. Find the Other Roots (con’t)
The resulting polynomial is a quadratic, but it doesn’t have real factors. Solve the quadratic set equal to zero by either using the quadratic formula, or by isolating the x and taking the square root of both sides.

40. Find the Other Roots (con’t)
The solutions to the quadratic equation:
For the polynomial:
The three complex roots of the polynomial are:

41. 4.5 Rational Functions Objective: In this lesson, we will:
Sketch graphs of rational functions
Look more in depth at the vertical & horizontal asymptotes
Find equations of rational functions

42. 4.5 Rational Functions and Their Graphs

43. Example Find the Domain of this Function. Solution:
The domain of this function is the set of all real numbers not equal to 3.

44. Arrow Notation Symbol Meaning x  a + x approaches a from the right.
x  a - x approaches a from the left.
x   x approaches infinity; that is, x increases without bound.
x  -  x approaches negative infinity; that is, x decreases without bound.

45. Definition of a Vertical Asymptote
The line x = a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a.
f (x)   as x  a f (x)   as x  a - f a y x
x = a f a y x
x = a
Thus, f (x)   or f(x)  -  as x approaches a from either the left or the right.

46. Definition of a Vertical Asymptote
The line x = a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a.
f (x)  as x  a + f (x)  -  as x  a - f a y x
x = a
x = a f a y x
Thus, f (x)   or f(x)  -  as x approaches a from either the left or the right.

47. Locating Vertical Asymptotes
If is a rational function in which
p(x) and q(x) have no common factors and a is a zero of q(x), the denominator, then x = a is a vertical asymptote of the graph of f.

48. Definition of a Horizontal Asymptote
The line y = b is a horizontal asymptote of the graph of a function f if f (x) approaches b as x increases or decreases without bound. f y x
y = b x y f
y = b f y x
y = b
f (x)  b as x   f (x)  b as x   f (x)  b as x 

49. Locating Horizontal Asymptotes
Let f be the rational function given by
The degree of the numerator is n. The degree of the denominator is m.
If n<m, the x-axis, or y=0, is the horizontal asymptote of the graph of f.
If n=m, the line y = an/bm is the horizontal asymptote of the graph of f.
If n>m,t he graph of f has no horizontal asymptote.

50. Strategy for Graphing a Rational Function
Suppose that where p(x) and q(x) are
polynomial functions with no common factors.
1. Find any vertical asymptote(s) by solving the equation q (x) = 0.
Find the horizontal asymptote (if there is one) using the rule for determining the horizontal asymptote of a rational function.
Use the information obtained from the calculators graph and sketch the graph labeling the asymptopes.

51. Sketch the graph of

52. The vertical asymptote is x = -2
The horizontal asymptote is y = 2/5

54. Section 4.5 Rational Functions
Find an equation of a rational function f that satisfies
x-intercept : 4
Vertical asymptote: x=-2
Horizontal asymptote: y= -3/5
hole at x=1

55. Section 4.5 Rational Functions
X-intercept -- x-4 must be factor in numerator (x-4)
Vertical asymptote-- x+2 is factor in denominator x-4
x+2
Horizontal asymptote--Mult. Numerator by -3 and denominator by 5
-3(x-4)
5(x+2)
Hole--x-1 must be factor in both num. & den.
f(x)= -3(x-4)(x-1)
5(x+2)(x-1)

56. Ch. 4 Review Answers