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
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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).