Partial Fractions, Method of Cancelling

[Ledge]

Homework Statement

Expand Using Partial Fractions:
f(x) = ( 5x-10 )/ ( (x+1)(x-4) )

Involves Integrating afterwards but I don't think this affects my method.

Homework Equations

This was a question in my mid semester exam, I got the answer correct but he insists my method is wrong.

The Attempt at a Solution

The method I used was to get into the form of:
( 5x-10 )/ ( (x+1)(x-4) ) = A/(x+1) + B/(x-4)
then, rearrange to:
5x-10 = A(x-4) + B(x+1)
and pick values of x to cancel the factors such as 4 and -1 and solve for A & B.
so:
5(4)-10 = B(4+1) → B = 2
5(-1)-10 = A(-1-4) → A = 3

I got this method from high school and have been using it ever since, I also found it clearly stated in the textbook we are currently using. After showing him he still insists the textbook and I are wrong. His reasoning is that finding A & B for a specific value of x cannot be assumed to work for all x. I don't understand his logic as A & B are constants.
Is there any way I can prove this to him? Like a rigorous proof of some sort. Or if I am possibly wrong, can someone explain it to me as he is not good with articulating at all.

Thanks in advance for any help :).

 

[vela]

Your method is fine. If the relation holds for all x, it should certainly hold for specific values of x, like x=4.

I'm not sure how you can convince your instructor, though.

 

[JHamm]

Isn't partial fractions included as a part of calculus classes?

 

[Ledge]

Yeah sorry it is, I thought it was assumed knowledge from high school and guessed it was algebra.

EDIT: Thanks vela, I guess I'll try to continue to get it through to him, if not I'll have to make a complaint or something.

 

[Mark44]

Ledge, I agree with you and vela. When you solve for the constants in a partial fractions decomposition, the equation you set up is an identity - one that holds for all values of x other than those that make any of the denominators vanish.

I sympathize with your frustration. I worked with a guy one time who insisted that all of his peer instructors at our college, and the textbook were doing things wrong. During my stint as the head of the math department, I tried to get him fired, but since he was a member of the union, all my efforts were in vain. Fortunately for the college and his students, he retired.

 

[HallsofIvy]

Oh, yeah, I know that type! When I was chair of a math department (yes, my shameful past) we had hired a guy that I soon found out had told Calculus students to find "max" and "min" values by just comparing the values of f(x) for integer x. For that and other reasons, we dismissed him after one semester. Fortunately, because he was a new hire, it was in his contract that the first semester was a "trial".

 

[stevenb]

Is there any way I can prove this to him? Like a rigorous proof of some sort. Or if I am possibly wrong, can someone explain it to me as he is not good with articulating at all.

Thanks in advance for any help :).

Your method can probably be directly tied to the method of residues shown in the following reference. The case of degenerate poles gets a little more tricky, but this is an accepted method.

http://en.wikipedia.org/wiki/Partial_fraction