1. Polynomial and Rational Functions
Chapter 2
Polynomial and Rational Functions

2. Section 1
Quadratic Functions

3. Quadratic Functions
Let a, b, and c be real number with a ≠ 0. The function f(x) = ax2 + bx = c is called a quadratic function.
The graph of a quadratic function is a special type of U-shaped curve that is called a parabola.
All parabolas are symmetric with respect to a line called the axis of symmetry, or simply the axis of the parabola.
The point where the axis intersects the parabola is called the vertex.

4. Quadratic Functions If a >0, then the graph opens upward.
If a < 0, then the graph opens downward.

5. The Standard Form of a Quadratic Function
The standard form of a quadratic functions
f(x) = a(x-h)2+ k, a ≠ 0
Vertex is (h, k)
|a| produces a vertical stretch or shrink
(x – h)2 represents a horizontal shift of h units
k represents a vertical shift of k units
Graph by finding the vertex and the x- intercepts

6. Vertex of a Parabola
The vertex of the graph f(x) = a(x)2+ bx + c is ( -b/2a, f(-b/2a)) EXAMPLE Find the vertex and x-intercepts -4x2 +x + 3

7. Polynomial Functions of Higher Degree
Section 2
Polynomial Functions of Higher Degree

8. Polynomial Functions
Let n be a nonnegative integer and let an, an-1, ….. …a2, a1, a0 be real numbers with an ≠ 0.
The function f(x) = anxn + an-1xn-1 +…… a2x2 + a1x + a0 is called a polynomial function of x with degree n.
EXAMPLE
f(x) = x3

9. Characteristics of Polynomial Functions
The graph is continuous.
The graph has only smooth rounded turns.

10. Sketching Power Functions
Polynomials with the simplest graphs are monomial of the form f(x) = xn and are referred to as power functions.
REMEMBER ODD and EVEN FUNCTIONS
Even : f(-x) = f(x) and symmetric to y-axis
Odd: f(-x) = - f(x) and symmetric to origin

11. Leading Coefficient Test
When n is odd: If the leading coefficient is positive (an >0), the graph falls to the left and rises to the right
When n is odd: If the leading coefficient is negative (an <0), the graph rises to the left and falls to the right

12. Leading Coefficient Test
When n is even: If the leading coefficient is positive (an >0), the graph rises to the left and right.
When n is even: If the leading coefficient is negative (an <0), the graph falls to the left and right

13. EXAMPLE
Identify the characteristics of the graphs
f(x) = -x3 + 4x
f(x) = -x4 - 5x2 + 4
f(x) = x5 - x

14. Real Zeros of Polynomial Functions
If f is a polynomial function and a is a real number, the following statements are equivalent.
x = a is a zero of the function f
x =a is a solution of the polynomial equation f(x)=0
(x-a) is a factor of the polynomial f(x)
(a,0) is an x-intercept of the graph of f

15. Repeated Zeros
A factor (x-a)k, k >0, yields a repeated zero x = a of multiplicity k.
If k is odd, the graph crosses the x-axis at x = a
If k is even, the graph touches the x-axis at x = a (it does not cross the x-axis)

16. EXAMPLE
Graph using leading coefficient test, finding the zeros and using test intervals
f(x) = 3x4 -4x3
f(x) = -2x3 + 6x2 – 4.5x
f(x) = x5 - x

17. Long Division of Polynomials
Section 3
Long Division of Polynomials

18. Long Division Algorithm
If f(x) and d(x) are polynomials such that d(x) ≠ 0, and the degree of d(x) is less than or equal to the degree of f(x), there exist unique polynomials q(x) and r(x) such that: f(x) = d(x) q(x) + r(x)

19. EXAMPLE Divide the following using long division. x3 -1 by x – 1
2x4 + 4x3 – 5x2 + 3x -2 by x2 +2x – 3
Remember to use zero coefficients for missing terms

20. Synthetic Division
Synthetic Division is simply a shortcut for long division, but you still need to use 0 for the coefficient of any missing terms.
EXAMPLE
Divide x4 – 10x2 – 2x +4 by x + 3
= x3 – 3x2 - x + 1 R 1

21. The Remainder and Factor Theorems
Remainder Theorem: If a polynomial f(x) is divided by x-k, then the remainder is r = f(k) EXAMPLE Evaluate f(x) = 3x3 + 8x2 + 5x – 7 at x = -2 Using synthetic division you get r = -9, therefore, f(-2) = -9

22. The Remainder and Factor Theorems
Factor Theorem: A polynomial f(x) has a factor (x-k) if and only if f(k) =0 EXAMPLE Show that (x-2) and ( x+3) are factors of f(x) = 2x4 + 7x3 -4x2 -27x – 18 Using synthetic division with x-2 and then again with x+3 you get f(x) = (x-2)(x+3)(2x+3)(x+1) implying 4 real zeros

23. Uses of the Remainder in Synthetic Division
The remainder r, obtained in the synthetic division of f(x) by x-k, provides the following information:
The remainder r gives the value of f at x=k. That is, r= f(k)
If r=0, (x-k) is a factor of f(x)
If r=0, (k,0) is an x-intercept of the graph of f

24. Section 4
Complex Numbers

25. The Imaginary Unit i
Because some quadratic equations have no real solutions, mathematicians created an expanded system of numbers using the imaginary unit i, defined as i = -1 i2 = -1 i3 = -i i4 = 1

26. Complex Numbers
The set of complex numbers is obtained by adding real numbers to real multiples of the imaginary unit. Each complex number can be written in the standard form a + bi . If b = 0, then a + bi = a is a real number. If b ≠ 0, the number a + bi is called an imaginary number. A number of the form bi, where b ≠ 0, is called a pure imaginary number.

27. Some Properties of Complex Numbers
a + bi = c+ di if and only if a=c and b=d.
(a + bi) + (c+ di) = (a +c) + (b + d)i
(a + bi) – (c+ di) = (a – c) + (b – d)i
– (a + bi) = – a – bi
(a + bi ) + (– a – bi) = 0 + 0i = 0

28. Complex Conjugates a + bi and a –bi are complex conjugates
(a + bi) (a –bi ) = a2 + b2
EXAMPLE
(4 – 3i) (4 + 3i) = = 25

29. Complex Solutions of Quadratic Equations
Principal Square Root of a Negative Number
If a is a positive number, the principal square root of the negative number –a is defined as
 – a = a i
EXAMPLE
 – 13 = 13i

30. TheFundamental Theorem of Algebra
Section 5
TheFundamental Theorem of Algebra

31. The Fundamental Theorem of Algebra
Linear Factorization Theorem
If f(x) is a polynomial of degree n, where n > 0, then f has at least one zero in the complex number system.
If f(x) is a polynomial of degree n, where n > 0, then f has precisely n linear factors. f(x) = an(x – c1)(x- c2)…(x–cn) Where c1,c2…cn are complex numbers

32. Find the zeros of the following: f(x) = x – 2 f(x) = x2 – 6x + 9
EXAMPLE
Find the zeros of the following:
f(x) = x – 2
f(x) = x2 – 6x + 9
f(x) = x3 + 4x
f(x) = x4 – 1

33. Rational Zero Test
If the polynomial f(x)= anxn + an-1xn-1 +…a2x2+a1x1 +a0 has integer coefficients, every rational zero of f has the form
Rational zero = p/q or constant term/leading coefficent
Where p and q have no common factors other than 1, and
p = a factor of the constant term a0
q = a factor of the leading coefficient an

34. EXAMPLE Find the rational zeros of f(x) = 2x3+3x2 – 8x + 3
Rational zeros p/q = ± 1, ± 3 / ± 1, ± 2
Possible rational zeros are ± 1, ± 3, ± ½, ± 3/2
Use synthetic division by trial and error to find a zero

35. Conjugate Pairs
Let f(x) be a polynomial function that has real coefficients. If a + bi, where b ≠ 0, is a zero of the function, the conjugate a – bi is also a zero of the function.
Rational zero = p/q
Where p and q have no common factors other than 1, and
p = a factor of the constant term a0
q = a factor of the leading coefficient an

36. EXAMPLE
Find a 4th degree polynomial function with real coefficients that has – 1, – 1, and 3i as zeros
Then
f(x) = a(x+1)(x+1)(x – 3i)(x+3i)
For simplicity let a = 1
Multiply the factors to find the answer.

37. EXAMPLE Find all the zeros of
f(x) = x4 – 3x3 + 6x2 + 2x – 60 where 1 + 3i is a zero
Knowing complex zeros occur in pairs, then 1 – 3i is a zero
Multiply (1+3i)(1 – 3i) = x2 – 2x +10 and use long division to find the other zeros of -2 and 3
x4 – 3x3 + 6x2 + 2x – 60/(x2 – 2x +10)

38. Find possible rational roots and use synthetic division
EXAMPLE
Find all the zeros of
f(x) = x5 + x3 + 2x2 – 12x +8
Find possible rational roots and use synthetic division

39. EXAMPLE
You are designing candle-making kits. Each kit will contain 25 cubic inches of candle wax and a mold for making a pyramid-shaped candle. You want the height of the candle to be 2 inches less than the length of each side of the candle’s square base. What should the dimension of your candle mold be? Remember V = 1/3Bh

40. Section 6
Rational Functions

41. Rational Function A rational function can be written in the form
f(x) = N(x)/D(x) where N(x) and D(x) are polynomials and D(x) is not the zero polynomial. Also, this sections assumes N(x) and D(x) have no common factors.
In general, the domain of a rational function of x includes all real numbers except x-values that make the denominator zero.

42. EXAMPLE
Find the domain of the following and explore the behavior of f near any excluded x-values (graph)
f(x) = 1/x
f(x) = 2/(x2 – 1) 2

43. Is essentially a line that a graph approaches but does not intersect.
ASYMPTOTE
Is essentially a line that a graph approaches but does not intersect.

44. Horizontal and Vertical Asymptotes
The line x= a is a vertical asymptote of the graph of f if
f(x) → ∞ or f(x) → – ∞
as x → a either from the right or from the left.
The line y= b is a horizontal asymptote of the graph of f if
f(x) → b
as x → ∞ or x → – ∞

45. Asymptotes of a Rational Function
Let f be the rational function given by
f(x) = N(x)/D(x) where N(x) and D(x) have no common factors then: anxn + an-1xn-1…./(bmxm +bm-1xm-1…)
The graph of f has vertical asymptotes at the zeros of D(x).
The graph of f has one or no horizontal asymptote determined by comparing the degrees of N(x) and D(x)
If n < m, the graph of f has the line y = 0 as a horizontal asymptote.
If n= m, the graph of f has the line y = an/bm as a horizontal asymptote.
If n>m, the graph of f has no horizontal asymptote

46. EXAMPLE
Find the horizontal and vertical asymptotes of the graph of each rational function.
f(x) = 2x/(x4 + 2x2 + 1)
f(x) = 2x2 /(x2 – 1)

47. Graphing Rational Functions
Let f be the rational function given by
f(x) = N(x)/D(x) where N(x) and D(x) have no common factors
Find and plot the y-intercept (if any) by evaluating f(0).
Find the zeros of the numerator (if any) by solving the equation N(x) =0 and plot the x-intercepts
Find the zeros of the denominator (if any) by solving the equation D(x) = 0, then sketch the vertical asymptotes
Find and sketch the horizontal asymptote (if any) using the rule for finding the horizontal asymptote of a rational function
Test for symmetry (mirror image)
Plot at least one point between and one point beyond each x- intercept and vertical asymptote
Use smooth cures to complete the graph between and beyond the vertical asymptotes

48. EXAMPLE Graph f(x) = 3/(x – 2) f(x) = (2x – 1)/x

49. Slant Asymptotes
Consider a rational function whose denominator is of degree 1 or greater. If the degree of the numerator is exactly one more than the degree of the denominator, the graph of the function has a slant (or oblique) asymptote.

50. EXAMPLE f(x) = (x2 – x) /( x+ 1) has a slant asymptote.
To find the equation of a slant asymptote, use long division.
You get x – 2 + 2/(x+1)
y = x – 2 because the remainder term approaches 0 as x increases or decreases without bound

51. EXAMPLE f(x) = (x2 – x – 2) /( x – 1) Find the x-intercepts
Find the y-intercepts
Vertical asymptotes
Slant asymptote
Try graphing using your calculator.

52. THE END