Write the partial fraction decomposition of the rational expression.

(80-14x)/(x^3 -8x^2 +16x)

(80-14x)/(x^3 -8x^2 +16x)

=(80-14x)/(x(x-4)²)
=A/x + B/(x-4) + C/(x-4)²

=>
A(x-4)²+Bx(x-4)+Cx = 80-14x
=>
x²(A+B) + (C-4B-8A)x + 16A = 80-14x
=>
A=5
B=-5
C=6
Can you take it from here?

I got it; thank you so much!

You're welcome!

To find the partial fraction decomposition of the rational expression (80-14x)/(x^3 - 8x^2 + 16x), we need to factor the denominator and express the given fraction as a sum of simpler fractions with these factors in the denominators.

First, let's factor the denominator:
x^3 - 8x^2 + 16x = x(x^2 - 8x + 16)

Now, we need to factor the quadratic term inside the parentheses, which is x^2 - 8x + 16. This quadratic can be factored as (x - 4)(x - 4), which simplifies to (x - 4)^2.

Therefore, the factored form of the denominator is: x(x - 4)^2.

Now, we can express the given rational expression (80-14x)/(x^3 - 8x^2 + 16x) as a partial fraction sum in the form of A/x + B/(x - 4) + C/(x - 4)^2.

Let's find the values of A, B, and C:

Multiplying both sides of the equation by the common denominator x(x - 4)^2:
(80 - 14x) = A(x - 4)^2 + Bx(x - 4) + Cx.

Now we can equate the coefficients of like terms on both sides:

1. For the constant term (the term without any x):
80 = A(-4)^2.

Simplifying this equation:
80 = 16A.

Thus, A = 80/16 = 5.

2. For the x term:
-14x = Bx(x - 4).

Expanding the right side:
-14x = B(x^2 - 4x).

Equating the coefficients on both sides:
-14 = B.

Thus, B = -14.

3. For the (x - 4)^2 term:
None of the terms on the right side have this term, so the coefficient is 0.

Therefore, C = 0.

So, the partial fraction decomposition of the given rational expression is:
(80-14x)/(x^3 - 8x^2 + 16x) = 5/x - 14/(x - 4) + 0/(x - 4)^2.

Simplifying the expression further, we get:
(80-14x)/(x^3 - 8x^2 + 16x) = 5/x - 14/(x - 4).