# Thread: Two Proofs (Partial Fractions and other)

1. ## Two Proofs (Partial Fractions and other)

Hi,

I am just curious about how to prove these following things:

Firstly, what's the proof for the following:

$
\frac{8x+7}{(2x+1)^2} = \frac{a}{(2x+1)^2} + \frac{b}{2x+1}$

I was also told that the polynomial in the numerator should be 1 degree lower than the one in the denominator - why?

Also, I was wondering.

$
y^2 = 3x - 12$
why does that become

$y = +\sqrt[2]{3x-12}\ or -\sqrt[2]{3x-12}$

Again, I'm looking for a proof explanation so that I can remember it and derive it if I forget.

Thanks!

2. Originally Posted by Rudey
Hi,

I am just curious about how to prove these following things:

Firstly, what's the proof for the following:

$
\frac{8x+7}{(2x+1)^2} = \frac{a}{(2x+1)^2} + \frac{b}{2x+1}$
What is the proof of what? Whether this equation is true or not depends upon what a and b are. Do you mean prove that there must be numbers a and b that satisfy that? For this particular problem, the best way to prove it would be to find the values of a and b. Or do you mean: "prove that for any $\frac{px+ q}{(rx+t)^2}= \frac{a}{rx+t}+ \frac{b}{(rx+t)^2}$"? Again, the best way to prove that would be to solve for a and b, in terms of p, q, r, and t.

I was also told that the polynomial in the numerator should be 1 degree lower than the one in the denominator - why?
I doubt that was what you were told- it's not true. You probably were told that the numerator must be at least one degree lower than the denominator. That's because if the degree of the numerator were greater than or equal to the degree of the denominator you could do a "long division" giving a polynomial plus a fraction in which the numerator does have degree less than the denominator.

Also, I was wondering.

$
y^2 = 3x - 12$
why does that become

$y = +\sqrt[2]{3x-12}\ or -\sqrt[2]{3x-12}$

Again, I'm looking for a proof explanation so that I can remember it and derive it if I forget.

Thanks!
I'm not sure what you want. $\sqrt{x}$ is defined (for real numbers) as the positive real number, y, such that $y^2= x$. Because -y (and no other number) will also have that property, $(-y)^2= y^2= x$, from $y= x^2$, it follows that $y= \sqrt{x}$ or $y= -\sqrt{x}$.