1. Mathematics Examples of Polynomials and Inequalities
Dr Viktor Fedun
Automatic Control and Systems Engineering, C09
Based on lectures by Dr Anthony Rossiter
Examples taken from the :
“Engineering Mathematics through Applications”
Kuldeep Singh Published by: Palgrave MacMillan
and

2. Example 1 (Page 145 Example 11)
[Mechanics]
The displacement, x(t), of a particle is given by:
x(t)= (t-3)2
Sketch the graph of displacement versus time
At what time(s) is x(t)=0?

3. Example 1 Solution Solution:
It is the same graph as the quadratic graph t2 but shifted to the right by 3 units.
x(t)=0 when t=3

4. Example 1 Solution
x(t)
x(t)= (t)2
x(t)= (t-3)2 t 3

5. Example 2 (Page 113 Exercise2 (c) q2
[Fluid Mechanics]
The velocity, v, of a fluid through a pipe is given by:
v = x2 – 9
Sketch the graph of v against x.

6. Example 2 Solution
v(x)
v(x)= x2
v(x)= x2-9 -3 3 x -9

7. Example 3 (Page 118 Exercise 2(d) q3)
[Electrical principles]
The voltage, V, of a circuit is defined as:
V = t2 – 5t + 6 (t ≥ 0)
Sketch the graph of V against t, indicating the minimum value of V

8. Example 3 Solution
In order to plot the graph, it helps to find the values of t for which the graph cuts the t axis, and the value of V for which the graph crosses the V axis.
For the t axis, factorising the polynomial function and then setting equal to zero will tell us of those values where the graph crosses the t axis (i.e. when v=0).
V = t2 – 5t + 6 (t ≥ 0)
=(t-2)(t-3)
So either (t-2)=0 or (t-3)=0 giving the crossings at t=2 and t=3
For the V axis, the graph crosses the V axis when t=0, giving V(t=0)=6

9. Example 3 Solution v(t) v(t)= t2-5t+6 6 2 3 t Note: t ≥ 02 3 t
Note: t ≥ 0
The minimum value is here

10. Polynomials
Functions made up of positive integer powers of a variable, for instance:

11. Degree of a polynomial The degree is the highest non-zero power

12. Typical names Degree of 0 constant Degree of 1 linear
Degree of 2 quadratic
Degree of 3 cubic
Degree of 4 quartic
Etc.

13. Multiplying polynomials
If you multiply a rth order by an mth order, the result has order r+m.
In general, you do not want to do this by hand, but you must be able to!
If you are not sure about multiplying out brackets, see me asap.

14. Factorising a polynomial
Discuss in groups and prepare some examples to share with the class.
What is a factor?
What is a factor of a polynomial?
What is the root of a polynomial?
What is the relationship between a factor and a root?
How many factors/roots are there?

15. Finding factors/roots
We factorise a polynomial be writing it as a product of 1st and/or 2nd order polynomials.

16. Finding factors/roots
We factorise a polynomial be writing it as a product of 1st and/or 2nd order polynomials.
Factors are numbers (expressions) you can multiply together to get another number (expressions):
Factors
2nd order polynomials are needed when this can not easily be expressed as the product of two 1st order polynomials.

17. Finding factors/roots
A roots is defined as the values of independent variable such that the function is zero. i.e.
‘a’ is a root of f(x) if f(a)=0.

18. Finding factors/roots
Find factors and roots is the same problem.
A factor (x-a) has a root at ‘a’.
If a polynomial has roots at 2,3,5, the polynomial is given as
`A` cannot be determined solely from the roots.
To factorise, first find the roots.

19. Problem Define polynomials with roots: -1, -2 , 3 4, 5,-6,-7
Find the roots of the following polynomials

20. What about quadratic factors
What are the roots of
How many roots does an nth order polynomial have?

21. What about quadratic factors
What are the roots of
How many roots does an nth order polynomial have?
Always n, but some are not real numbers.

22. Solving for the roots with a clue
Find the roots of

23. Solving for the roots with a clue
Find the roots of
By inspection, one can see that w=-1 is a root.

24. Solving for the roots with a clue
Find the roots of
By inspection, one can see that w=-1 is a root. Therefore extract this factor, i.e.
Hence, by inspection, A=1, B=2, C=1

25. Solving for the roots with a clue
Find the roots of
Given this quadratic factor, we can solve for the remain two roots.
Hence, there are 3 roots at -1.

26. For the class
Solve for the roots of the following.

27. Sketching polynomials
Sketch the following polynomials.
Key points to use are:
Roots (intercept with horizontal axis).
If order is even, increases to infinity for +ve and –ve argument beyond domain of roots.
If order is odd, one asymptote is + infinity and the other is - infinity.

28. Why are polynomials so important?
Within systems engineering, behaviour is often reduced to solving for the roots of a polynomial. Roots at (-a,-b) imply behaviour of the form:
You must design the polynomial to have the correct roots and hence to get the desired behaviour from a system.

29. Inequalities We will deal with equations that involve the symbols.
A key skill will be the rearrangement of functions.
What do these symbols mean?
Discuss in class for 2 minutes.

30. Which of the following are true?

31. Changing the order
In the following replace > by < or vice versa.

32. Linear Inequalities

33. Linear Inequalities

34. Linear Inequalities

35. Linear Inequalities

36. Linear Inequalities

37. Linear Inequalities
Example

38. Linear Inequalities
Example

39. Linear Inequalities
Example or

40. Polynomial Inequalities
Example

41. Polynomial Inequalities Example
Recipe
1. Get a zero on one side of the inequality

42. Polynomial Inequalities Example
Recipe
1. Get a zero on one side of the inequality
2. If possible, factor the polynomial

43. Linear Inequalities
Example

44. Polynomial Inequalities Example
Recipe
1. Get a zero on one side of the inequality
2. If possible, factor the polynomial
3. Determine where the polynomial is zero

45. Polynomial Inequalities Example
Recipe
1. Get a zero on one side of the inequality
2. If possible, factor the polynomial
3. Determine where the polynomial is zero
4. Graph the points where the polynomial is zero

46. Polynomial Inequalities Example
Recipe
4. Graph the points where the polynomial is zero

47. Polynomial Inequalities Example
Recipe
4. Graph the points where the polynomial is zero

48. Polynomial Inequalities
For the class

49. Rational Inequalities

50. Rational Inequalities

51. Rational Inequalities

52. Rational Inequalities

53. Rational Inequalities
For the class

54. Rational Inequalities
For the class

55. Rational Inequalities
For the class

56. Rational Inequalities
For the class

57. Rational Inequalities
For the class

58. Absolute Value Equations

59. Absolute Value Equations

60. Absolute Value Equations

61. Absolute Value Equations

62. Absolute Value Equations

63. Absolute Value Inequalities

64. Absolute Value Inequalities

65. Absolute Value Inequalities

66. Absolute Value Inequalities

67. Absolute Value Inequalities

68. Absolute Value Inequalities

69. Absolute Value Inequalities

70. Absolute Value Inequalities