1. College Algebra
Acosta/Karowowski

2. Polynomial and rational functions
Chapter 7

3. Polynomial functions
Chapter 7 – Section 1

4. Review and definition f(x) = x g(x) = x2 k(x) = x3 m(x) = x4
etc. Called power functions –
a polynomial function is a sum of power functions where the exponents are whole numbers (ie – no fractions – no negative in exponents)

5. Which of the following are polynomial
f(x) = 3x3 – 5x + 8
𝑔 𝑥 = 3𝑥 −5 +4 𝑥 3
L(x) = 3x – 7
𝑘 𝑥 = 6 𝑥 3 − 2 3 𝑥 7
𝑚 𝑥 = 6𝑥 +5𝑥

6. Other forms of polynomial functions
Multiplying, adding, or subtracting polynomials always results in a polynomial
Therefore polynomial functions may appear factored or as sums
g(x) = (x – 9)3(x + 2)2
m(x) = (3x + 5)4 – (2x2 + 3)3

7. Degree of a polynomial/leading coeffiecient
When the polynomial is simplified and written in descending order the degree of the polynomial is the degree of the leading term (in one variable polynomials this is simply the exponent of the leading term)
Ex: g(x) = 3x5 – 2x3 + 18x2 + 10
It is neither necessary nor desirable to simplify a factored polynomial in order to determine its degree
f(x) = (x – 2)2(x – 5)3(x + 2)
k(x) = (4x2 – 7)2 (x3 + 6)
m(x) = (3x + 5)4 – (2x2 + 3)3

8. Analyzing polynomial functions- what the equation (or the graph) tells us
Domain
Range
Maximums, minimums -
end behavior
y- intercept
x – intercept
Turning points – local/relative maximums and minimums
General shape – end behavior (limits)

9. Domain
There are no restrictions on power functions, multiplication, addition or subtraction so there are NO restrictions on polynomial functions
Conclusion - domain for ALL polynomial functions is all real numbers

10. Range/ max -min
Since the domain is unrestricted the question of range translates into “Is there a maximum or minimum value?”
With power functions – even powers attain a maximum or minimum while odd powers do not
What happens when we add/subtract even and odd power?

11. Conclusion Range is determined by degree of function
even degree has a restricted range. It may not be easy to find but it is known to exist - if leading coefficient > 0 (pos)the range has a minimum – If the leading coefficient < 0 (neg) it flips the polynomial and the range has a maximum
Odd degree functions have a range of all real numbers – the function goes
neg. infinity on the left to positive infinity on the right. Essentially an
increasing function
Leading coefficient < 0 (neg) flips the polynomial making it go from
positive infinity on the left to neg infinity on the right – essentially a
decreasing function.
The above statements are sometimes referred to as “End behavior”. This behavior is determined by degree and by the leading coefficient

12. Examples
Determine degree for the following polynomials – what does this tell you about the graph?
f(x) = 2x5 – 7x4 + 3x2 – 9
g(x) = (x + 2)(x – 5)( x + 3)2
k(x) = (x + 2)3 – (x + 3)2 + 5x – 7
m(x) = (x – 2)(3 – x) (2x – 5)2
z(x) = -3x7 – 4x5 + 2x3
Determining max/min cannot be done with algebra except by trial and error (making a table) this is not a desirable method and Calculus offers a way to find max and min points

13. Using a graphing calculator to estimate local max/min
Ex x x x x+45
Ex: (x – 3)(x – 6)(x + 4)
[-8,8,1] x[-300,200,50]

14. Y- intercepts f(0) = y determines the y – intercept
Ex: f(x) = 2x5 – 7x4 + 3x2 – 9
g(x) = (x + 2)(x – 5)( x + 3)2
k(x) = (x + 2)3 – (x + 3)2 + 5x - 7

15. X- intercepts- what’s happening in the middle of the graph
Given f(x) = y , Solving f(x) = 0 determines the x-intercepts
The solutions to this equation are often referred to as the zeroes of the function and by association the x – intercepts are also sometimes called the zeroes of the function
One method of solving a polynomial equation is by factoring
solutions x - intercepts
factors
Conclusion: given f(x)
x = a is a solution of f(x) = 0 if and only if (x –a) is a factor of f(x) and if and only if (a,0) is a point on the graph.

16. examples
Determine the degree of each polynomial and the x-intercepts of the graph and sketch the graph
g(x) = (x – 2)(x + 4)(x – 9)(5x + 6)
k(x) = .1x(x + 5)(2x – 9)

17. Example Find all solutions to x3 – 64 = 0
sketch the graph for f(x) = x3 – 64
Find all solutions to x4 – 10x2 – 24 = 0
sketch the graph for g(x)= x4 – 10x2 – 24
Find all solutions to x5 – 9x3 + 8x2 – 72 = 0
sketch the graph for k(x) = x5 – 9x3 + 8x2 – 72

18. Multiplicity and x-intercept behavior
When a solution occurs more than once it is said to “have multiplicity”
(x – 5)2 yields a solution of 5 with multiplicity of 2 – the x-intercept (5,0) is said to have multiplicity of 2
(x - 5)7 solution of 5 with multiplicity of 7
The behavior of these intercepts corresponds to what we already know about power graphs

19. Multiplicity of factors
j(x) = (-1/512)(x + 8)3(x – 7)2(x + 2)
Odd multiplicity –
Even multiplicity

20. Ex: Mult/intercepts Sketch a graph of the following polynomials
f(x) = (x + 2)(x – 3)(x – 7)
g(x) = (x + 2)2(x – 3)3 (x – 7)
k(x) = -2(x + 2)(x – 3)(x – 7)2

21. Using long division to factor
The factors of a polynomial are not always easy to find –
Pertinent fact about factors :
If a number is a factor of another number then it divides with no remainder

22. Long division/synthetic division
You can always do long division
You can only do synthetic division when you are dividing by a binomial of degree 1 with a coefficient of 1

23. Synthetic division
It’s a lot of writing to write all the variables over and over again. Synthetic division is a “shorthand” for long division –
(2x7 + 6x6 - 9x4 - 2x3 + 3x + 6)/ (x + 2) becomes:
(notice the sign change and that every term is represented)
-2|
Answer:

24. Example (x3 - 4x2 – 7x + 10)÷ (x + 2)
(6x4 +10x3 – 19x2 – 25x +10)/(2x2 – 5)

25. Examples (x3 – 5x2 + 7x -6)÷ (x – 4) (x4 – 2x2 + 5x + 5)/ (x + 2)

26. Division/factoring/zeroes(solutions)
Importance :
if the remainder is zero then (x – a) is a FACTOR of the polynomial
ie : we can use division to find factors that are not binomial factors
Given x + 2 is a factor of the function p(x) = x3 - 4x2 – 7x + 10
then
(x3 - 4x2 – 7x + 10)÷ (x + 2) = x2 - 6x +5
And x2 - 6x +5 is also a factor - therefore the factors of the polynomial are (x +2)(x -5)(x -1)
If there are enough binomial factors we can determine ALL the solutions (rational, irrational, and imaginary) for the given polynomial

27. Example Given g(x) = X5 + 2x4 – 12x3 – 24x2 + 27x + 54
is x = -2 a solution?
is x = 1 a solution?
is (x + 3 ) a factor of g(x)?
is (x +1) a factor of g(x)?

28. Rational factor theorem- what factors are possible
1) like the ac method of factoring a quadratic - any rational solution is determined by the factors of the leading coefficient and constant terms
axn +………….+ c
has rational solutions of 𝑐 1 𝑎 where c1 and a1 are
factors of c and a
2) any rational factors yield a solution to the equation
f(x) = 0
Thus (x – a) is a factor if f(a) = 0
3) any factor will divide nicely into the polynomial – thus yielding other factors

29. Example: x – intercepts vs solutions
Ex: Given f(x) = 2x3 – x2 – 7x + 6
First - possible solutions : 1,2,3,6
2: 1,2
there are at most 16 possible solutions
Second - determine at least one actual solution
f(1) f(-1) f(2) f(-2) ……
Third – divide the polynomial by (x ) and factor the resulting polynomial
solutions are :
x- intercepts are:

30. Examples Find all solutions for 3x3 - 5x2 - 34x – 24 = 0
Find all x- intercepts for g(x)= 3x3 - 5x2 - 34x – 24
sketch the graph
Find all solutions for x4 + 2x3 – 11x2 + 8x = 60
Find all x- intercepts for k(x) = x4 + 2x3 – 11x2 + 8x-60

31. Summary domain/ range - (end behavior) y – intercepts
x-intercepts - factors - multiplicity
finding possible factors by ac rule
testing to see if a binomial is a factor
finding factors by division
Use the above information to sketch a graph

32. Assignment
P 7 -23(1-10odd)( odd)(37-59 odd- use tracing on the word problems)(64 )(65 – 70 odd)
worksheet

33. Rational functions
Chapter 7 – section 2

34. Rational functions
A rational function is the quotient of 2 polynomial functions
𝑓 𝑥 = 𝑝(𝑥) 𝑞(𝑥) where p(x) and q(x) are
polynomial functions
Ex:

35. Domain
Domain of rational function are generally restricted - Denominator cannot equal zero
This has already been discussed
Review: find domain for
𝑓 𝑥 = 𝑥 2 −4 𝑥 2 −7𝑥− 𝑔 𝑥 = 9 𝑥 2 − 6𝑥 + 4
j x = 𝑥 −5 𝑥 2 +9
Range may or may not be restricted – this is not easily determined and will not be discussed here

36. Key elements vertical asymptotes and “holes”
A restriction in the domain means that there are NO points above those x-values - since these are single points (not intervals) the graph must “skip” that point.
This either creates a “hole” or a “wall” (called an asymptote)
Horizontal asymptotes: these occur when the quotient approaches a constant value for large values of x - they frequently cause a skip in the range but not always
Oblique asymptotes

37. Vertical asymptotes –vs- holes
If a factor in the denominator also occurs in the numerator the restriction at that value creates a hole
Otherwise a restriction causes an asymptote
Examples:
𝑓 𝑥 = 𝑥 2 −4 𝑥 2 −7𝑥−18
𝑔 𝑥 = 3𝑥 −7 𝑥 2 −2𝑥 −24

38. Horizontal asymptotes – end behavior
Recall what happens when x is HUGE either negative or positive
The behavior was dependent on the degree of the polynomial
Now we are dividing – there are 3 possibilities

39. Conclusions Given 𝑓 𝑥 = 𝑝(𝑥) 𝑞(𝑥)
where the degree of p(x) = a and the degree of q(x) = b
If a < b the horizontal asymptote is y = 0
If a> b there is no horizontal asymptote (there is a possibility of a slant asymptote)
If a = b the horizontal asymptote is the fraction created by the leading coefficients of numerator and denominator

40. examples
𝑓 𝑥 = 𝑥+5 3𝑥 2 −8𝑥
𝑘 𝑥 = 5 𝑥 2 −9 𝑥+2
𝑔 𝑥 = 7𝑥 −15 2𝑥 −6

41. Oblique asymptotes/ slant asymptotes
When the numerator is the higher degree
𝑝(𝑥) 𝑞(𝑥) =𝑓 𝑥 + 𝑟(𝑥) 𝑞(𝑥) where f(x) is a polynomial function and r(x) is the remainder from dividing.
f(x) defines the end behavior of the original function and if f(x) is linear is called a slant asymptote

42. examples
𝑘 𝑥 = 2 𝑥 4 −5 𝑥 2 2 𝑥 3 +5𝑥
6 𝑥 𝑥 4 −2𝑥 −15 3 𝑥 3 +4 𝑥 2

43. Rational graphs
From previous chapters you should know what the graph of f x = 1 𝑥 looks like and any transformations on that graph
Beyond that you should recognize a rational graph as a rational graph as opposed to a polynomial graph when you see one

44. Assignment
P616(1-16) (28 – 32)