Proving the Remainder Theorem

Okay folks, it's time that you put your thinking caps on. We observed some interesting results in class today when it came to dividing polynomial functions by expressions that were not generated by the roots of the functions. To prove these results to be true for every case, we will have to remember a few things about division, and especially about remainders.

When dividing a number, say 79, by another number, say 7, we get a remainder of  2. If I were to write this statement differently then I could say that 79 = (7)(11) + 2  (you should try out and verify what I am saying here).

Now see if this same kind of statement works for an actual example of a polynomial that is divided by an expression that is not a factor. You can use the two examples from class and try and write the statement I showed you but instead using the polynomials.

After this the work gets interesting and tricky. Try and write a similar statement for a random polynomial f(x) being divided by (x-a) and you can call the remainder r. bring to class anything and everything that occurs to you


NOTE: The class work that you have to complete are questions 1 and 2 from page 90 of your text book.

Homework due Friday, 17/02/12

Page 76, Ex. 4.3

#1 (b),1 (d),1 (e),1 (j), 3, 5, and 9.

 

Remember that a solution set is the same thing as saying the value(s) of x (or whatever input variable) that satisfy something. Please don't try to do these algebraically; visualize the function by first making the inequality a roots argument and then see what values satisfy the statement. For sketching the function you have options:

1. Find roots and see if the function is opening up or down.

OR

2. Convert to a form where the transformations are easy to see and then sketch it (you may still need to find roots for the solution set to the inequality).

Either way, roots are important!

Homework due Monday, 13/02/12

Page 74-75, Ex. 4.2

#10, 12, 14, 18 and 19

Remember, the first 3 questions are along the same lines as the previous ones you have done for homework. However, 18 and 19 deal with the intersections of linear and quadratic functions. Now, here's a clue: You will be making use of the discriminant in these questions as well.

Homework due Friday, 10/02/12

Page 74, Ex. 4.2

# 5, 7 and 9.

Homework due Monday, 06/02/12

Page 67, Ex. 4.1
#14

Page 73-74, Ex. 4.2

#1, 2, 3 and 4

Remember, when they are talking about the "nature" of the roots, they mean whether there is one real root, two real roots or no real roots. The discriminant helps you determine exactly this when the function is in standard form. You should certainly post your queries about the homework on the blog. Some of you may be wondering why they are using the word "real" to describe the roots every time. Do you remember that we learned about the Real numbers? Think about it and we can discuss this further in class.

Also, don't forget that your test on quadratics is on Tuesday, 7th February, 2012.