In the study of this course, in my opinion, polynomial and quadratic equations have been the most practical hence valuable to me both now and in my future day to day life. Although it may not be obvious for most of us, but it is a fact that people in all kinds of professions, from mathematicians, engineers to meteorologists use polynomials and quadratic equations to solve real life problems in their fields (Sturmfels,2002). However, polynomial equations can at times provide limited information but they prove useful in intricate analysis and in retrieving of additional information (Sturmfels,2002). For example,
Suppose you want to fence a field that is rectangular in shape. Say for some reason, maybe due to budget constraints, we want the field to have an enclosed area of 75 ft2. We also want the width of this field to be 3 feet longer than the length of the field. How can we calculate the field’s dimension (Paul’s Online Math Notes, 2003)?
Solution
If we let x be the length of the field thus length= x and the width will be . Area of a rectangle is given by finding the value of the length times width, therefore
Now, this is a quadratic equation which can be rewritten as:
Using the quadratic formula gives two answers, one positive and the other negative but since length cannot be negative, we go with the positive one, hence
Therefore, the length of the field is 7.2892 feet. The width is 3 feet longer than this and so is 10.2892 feet.
Question 2
In my opinion, the most important information that will be of use to me is how to form polynomial equations of different degrees and hence be able to create models and solve problems in different fields, for example, since polynomials are used regularly to describe curves of different kinds, they are used in the real world in curving graphs, a good example of this is how engineers who design roller coaster use polynomials to describe the curves in all their rides (Karim, 2006). However, the use of very complex, high degree polynomial equations may be of no value to me since I do not think i will be handling such complex mathematics.
Question 3
• Can you think of one real-world example of when the concept of functions might be useful? Do you think you will ever use functions in your life to solve problems? If yes, explain how and why; if no, explain why not.
Answer 3: With Examples
Functions are important tools in mathematics used whenever one variable, say y depends on another say x. For example, The area of a circle depends on its diameter/radius. A better real life yet simple example of how a function works is the amount of money your mother makes per month depends on the number of hours she works in the same month (Quadratic Function Models, 2010), i.e income is a function of time. If she is being paid $100 per hour and she works 50 hours a month then we can calculate her monthly income easily. We use functions in our day to day living even if we do not notice it sometimes, for instance, considering a snack, stamp or soda machine where you punch a specific button and a specific product comes out. In this machine, the function rule is the product price, the input is both the money and the punched button and the output is the product and in case more money was inserted then the change also counts as output. Therefore functions are basically statements that “if you tell me this variable, i can tell you the other one” (Quadratic Function Models, 2010).
Question 4
In this course, there are a number of concepts that i found really easy to grasp yet some were quite challenging. The ones that were easiest for me to grasp include simplifying polynomials, using the distribution property with polynomials and performing polynomial operations, this because the process of both simplifying and performing polynomial operations is logical in the steps that it follows and with a little care, you can avoid any errors. On the other hand, solving quadratic equations by graphical method and the use of the quadratic formula was most challenging. This i can associate with the fact that i found the formula hard to grasp and sometimes interpret and in use of graphs, sometimes the answers are not clear, for example representing a polynomial such as f(x)= 3×4-5×3+7×2-4x+2 is quite challenging.
Question 5
I would tell him that i have learnt what polynomial equations are i.e. algebraic expressions which add constants and variables. The variables have co-efficients which multiply the variables, and the variables may be raised to powers of non-negative integers hence defining the polynomial’s degree. For example, a quadratic expression is a polynomial of order 2 while a polynomial of order one is called a linear expression. In this course, i’ve also learnt how to take notes efficiently and effectively, how to pace myself with homework, engage in better studying habits such as reading in time and even before lectures and also how to manage my study time better.
Paul’s Online Math Notes, (2003) http://tutorial.math.lamar.edu/Classes/Alg/QuadraticApps.aspx
(Sturmfels, B. (2002) Solving Systems of Polynomial Equations Department of Mathematics, University of California at Berkeley.
Quadratic Function Models (2010) http://dufu.math.ncu.edu.tw/calculus/calculus_pre/node6.html. Retrieved on 14-11-2012
Karim, N, A. (2006). Understanding Quadratic Functions Using Real World Problems and IT. http://math.unipa.it/~grim/21_project/Karim291-295.pdf
