(7x^2-16)/(x^2-2x)

solve by using partial fraction decomposition. i cant figure out how to do this because after i split up the bottom the A and B don't end up having an x^2 in order to equal the numerator of the original fraction. so confused!

The problem here is that the partial expansion looks like:

A + B/(x-2) + C/x

To avoid the problem of the x^2 in the top, let's do a division first.

By long division ....
(7x^ - 16)/(x^2 - 2x)
= 7 + (14x-16)/(x^2 - 2x)

so now let's just work on that last term.

let (14x -16)/(x^ - 2x) = A/x + B/(x-2)
= (A(x-2) + Bx) / (x(x-2))

then 14x - 16 = A(x-2) + Bx
let x = 0
-16 = -2A
A = 8

let x = 2
12 = 2B
B = 6

so (7x^2-16)/(x^2-2x) = 7 + 8/x + 6/(x-2)

R(x) = (7x^2-16)/(x^2-2x) =

(7 x^2 - 16)/[x(x-2)]

Near x = 0, the singular behavior is:

R(x) = 1/x (7 x^2 - 16)/(x-2) =

1/x (-16/(-2)) + non-singular terms =

8/x + non-singular terms.

Near x = 2, the singular behavior is:

R(x) = 1/(x-2) (7 x^2 - 16)/x =

1/(x-2) [7*4-16]/2 + non-singular terms

=

6/(x-2) + non-singular terms.

R(x) minus alll singular terms =

R(x) - 8/x - 6/(x-2)

Now a rational function from which we have subtracted all singular terms doesn't ave any singularities, so it is actually a polynomial. We can find this polynomial by considering the behavior of the function at infinity. We see that the limit of x to infinity of R(x) exists and is equal to 7. The singular terms all go to zero at infinity. So,
R(x) minus the singular terms, which we know to be polynomial tends to zero at infinity, therefore the polynomial is a
cosntant function equal to 7 everwhere.

We can thus conclude that:

R(x) - 8/x - 6/(x-2) = 7 -------->

R(x) = 7 + 8/x + 6/(x-2)

To solve the given rational expression using partial fraction decomposition, let's go step by step:

Step 1: Factor the denominator
Start by factoring the denominator, x^2 - 2x. This can be factored as x(x - 2).

Step 2: Write the partial fraction decomposition
The next step is to express the given rational expression as a sum of partial fractions. In this case, we need to split the rational expression in terms of the factors of the denominator. Since the denominator has two distinct linear factors, we will write the partial fraction decomposition as:

(7x^2 - 16) / (x^2 - 2x) = A/x + B/(x - 2)

Step 3: Clear the denominators
To solve for the coefficients A and B, we need to eliminate the denominators. Multiply both sides of the equation by (x^2 - 2x):

(7x^2 - 16) = A(x - 2) + B(x)

Step 4: Expand and collect like terms
Expand the right side of the equation and collect like terms:

7x^2 - 16 = Ax - 2A + Bx

Now, group the like terms:

(7x^2 + Bx) - (2A) - 16 = 0

Step 5: Equate the coefficients
Since the left side of the equation must equal zero, we can equate the coefficients of the like powers of x on both sides. In this case, we have x^2 terms and a constant term. Setting the coefficients equal to zero, we get two equations:

For x^2 term: 7 = 0
This equation has no solution, which means there is no coefficient for the x^2 term in the partial fraction decomposition.

For x term: 0 = A + B
This equation tells us that the coefficient of x in the numerator of the original expression is equal to the sum of the coefficients A and B.

For the constant term: -16 = -2A
This equation helps us find the value of A.

Solving -2A = -16, we find A = 8.

Using A = 8 in the equation A + B = 0, we find B = -8.

Step 6: Write the partial fraction decomposition
Substituting the values of A and B into the partial fraction decomposition, we get:

(7x^2 - 16) / (x^2 - 2x) = 8/x - 8/(x - 2)

Therefore, the solution to the given rational expression using partial fraction decomposition is 8/x - 8/(x - 2).