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Algebra (all content)

Course: Algebra (all content)>Unit 13

Lesson 13: Partial fraction expansion

Intro to partial fraction expansion

Sal explains what partial fraction expansion is by rewriting (x²-2x-37)/(x²-3x-40) as the sum of 1 and two rational expressions with linear denominators. Created by Sal Khan.

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• I'm confused how he can just substitute in 8 and -5 for x. I get why but not how that works. Doesnt that mean that the expanded fraction only applies to when x is equal to 5 or -8? But thats indeterminate.... I don't know
• Unlike a system of equations where there are only unknown constants as variables, this kind of problem has unknown constants (what you're solving for) and a independent variable. Because of this, the statement must be true for all x. Thus, if it works for one specific x, it works for all x. We choose those two particular x values because they simplify the math a lot.
• How do we know that a decomposed part of the fraction (the A or B to solve for) is going to be a constant term? I understand the general method of decomposition, but why can we assume that?
• THe degree of the denominator always has to be bigger than the degree of the numerator. Therefore, if there is a LINEAR term (ax+b) in the denominator, then the numerator has to be a constant (A, B, C, whatever you wanna call it).
If there is a QUADRATIC term in the denominator (ax^2 +bx+c), then the numerator can be either LINEAR OR CONSTANT, since a quadratic equation has degree 2. When you do this in a decomposition, then you put Bx+C in the denominator (or whatever variables you wanna use). If the numerator ends up being a constant, then B=0.
By the way, something like (x+a)^2 counts as LINEAR.
Hope that helped :)
• The question is i have, why does the degree on the numerator needs to be lower than the degree of the denominator? can someone give an explanation? Thanks
• The degree of the numerator doesn't necessarily have to be lower than the degree of the denominator. But if we make it so, the work would be much more easier. Lets say for example: If the degree is more in the numerator , then instead of only adding constants , variables should also be added like A+Bx in the numerator. Too many unknowns=too much confusion
(1 vote)
• What is the purpose of partial fraction expansion? In what areas of math does this apply and facilitate?
• This is good for calculus, and differential equations.
• How do you know which factor to put under the letter A. What would've happened if you put A/(x-8)+B/(x+5) instead?
• Does that mean A/(x-8)+B/(x+5) is equivalent to A/(x+5)+B/(x-8)?
• Why is it allowed to set x to different arbitrary values?
• x is a variable; you can set it to be any arbitrary value you feel like. It's important to note that he would have arrived at the same values for A and B had he chosen some other, less convenient values for x as well.
• What happens if the denominator has a factor which is also present in the numerator?Should the factors be cancelled?
• Hello there, if you mean by factors then you can cancel. However, do not mix factors with terms such as
(5x)/(x) (you can cancel) but (5+x)/x. in this you cannot cancel the x
• The main problem where I see a real leap to something new is in the video. (x+5)(x-8) becomes "magically" (x+5) +(x+8). This is very confusing and there is no previous video that I can find where this is ever done.
• I take it you meant that (x+5)(x-8) "becomes" (x+5) + (x-8) ?

That's not what is going on. Those are the denominators he has decided to use, that is what this video is about, and he has to go through the steps outlined in the rest of the video to make that happen for his chosen denominators.