Partial fraction problem?

Question

Suppose, we want to convert "x^2/((x+1)(x+2)^2)" to partial fraction.Then we write:
x^2/((x+1)(x+2)^2)= A/(x+1) +B/(x+2) +C/(x+2)^2.
But my question is,why we write " B/(x+2)"? Why not "B/(x+2)^2?"
please tell me reasons.

Answer 1

Let's try it and see what the problem is.
Suppose we want
A/(x+1)+B/(x+2)^2=x^2/(x+1)(x+2)^2.

==>A(x+2)^2+B(x+1)=x^2

A(x^2+4x+4)+B(x+1)=x^2

==>A=1

==>4+B=0
B=-4
and everything works out.

But suppose instead you had

(X^2+1)/(X+1)(X+2)^2

Then as before you'd need

A(X+2)^2+B(X+1)=X^2+1

A(X^2+4X+4)+B(X+1)=X^2+1==>A=1 to get the X^2
==>B=-4 to cancel the 4X
but then there is no constant term, so the expansion is invalid.
In general you need three terms to get the three coefficients of the quadratic to match up properly.

Answer 2

It's somewhat difficult to explain, but here goes:

Because B is a number, instead of having:

B/(x+2), we essentially have B^2/(x+2)^2.

This means that B would be limited to only being represented by its squares. So unless a number was the square of something else (ie. 1, 4, 9, 16 etc), it could not be in the numerator for that second term.

By having a linear term AND a square term (ie b/x+2 and c/(c+2)^2), we're able to use more numbers (than just the squares) to represent the original fraction.

Answer 3

Suppose we wrote :

x*2/((x+1)(x+2)*2) = A/(x+1)+B/(x+2)*2+C/(x+2)*2.

Then, combining the last two terms on the right side of the equation, we get:

x*2/((x+1)(x+2)*2) = A/(x+1) + (B+C)/(x+2)*2.

As a result, we would be solving the equation for only 2 variables,
A, and B+C, which we can relabel as B'=B+C.

The original problem required a solution of three distinct variables, A,B,and C.

This should be some help in answering your question.

Answer 4

the question is good.the answer is 1/2+1/2^2=3/2^2
so if you are asked to reduce 3/2^2 into partial fractions it could be 1/2+1/2^2 isn't it?
3/(x+1)^2=A/(x+1)+B/(x+1)^2 as the LCD of both (x+1)^2 and (x+1) is (x+1)^2 as (x+1) is a factor of (x+1)^2

Answer 5

one million) element the denominator. 7x - 4 / x^2-x-12 7x - 4 / ((x - 4)(x + 3)) 2) Set it equivalent to A/f_1(x) + B/f_2(x). 7x - 4 / ((x - 4)(x + 3)) = A / (x - 4) + B / (x + 3) 3) Multiply the denominator to the two facets. 7x - 4 = A(x + 3) + B(x - 4) 4) Plug in numbers for x to sparkling up for a particular answer of A and B. x?-3, 7(-3) - 4 = A(-3 + 3) + B(-3 - 4) -21 - 4 = A(0) + B(-7) -25 = -7b b = 25 / 7 x?4, 7(4) - 4 = A(4 + 3) + B(4 - 4) 28 - 4 = A(7) + B(0) 24 = 7A A = 24 / 7 5) Now plug A and B back into the unique equation. (A, B) ? (24 / 7, 25 / 7), A / (x - 4) + B / (x + 3) (24 / 7) / (x - 4) + (25 / 7) / (x + 3) 6) answer. 7x - 4 / x^2-x-12 = 24/(7x - 28) + 25/(7x + 21)

Answer 6

Because you want to break the denominator into its linear terms and (x+2)^2 is not linear it is an x^2 term. Of course you know that (x+2)^2 = (x+2)(x+2) ?