How would you write the partial fraction decomposition of the rational expresion 1/(x^2)-25?
I know that you factor the denominator, so that's 1/(x-5)(x+5). What are the next steps?
To write the partial fraction decomposition of the rational expression 1/(x^2 - 25), after factoring the denominator as (x - 5)(x + 5), you need to determine the unknown coefficients A and B in the following form:
1/(x^2 - 25) = A/(x - 5) + B/(x + 5)
To find the values of A and B, you can use a common method called the "method of partial fractions." Here are the steps to follow:
1. Set up the equation: 1/(x^2 - 25) = A/(x - 5) + B/(x + 5)
2. Multiply both sides of the equation by (x^2 - 25) to clear the denominators:
1 = A(x + 5) + B(x - 5)
3. Simplify the equation:
1 = Ax + 5A + Bx - 5B
4. Combine like terms:
1 = (A + B)x + (5A - 5B)
5. Since two polynomials are equal, their corresponding coefficients must be equal. Therefore, we have the following system of equations:
A + B = 0 (coefficient of x)
5A - 5B = 1 (constant term)
6. Solve the system of equations. In this case, we can solve for A by multiplying equation 1 by 5, and then subtracting it from equation 2:
5A - 5B = 1
-(5A + 5B = 0)
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-10B = 1
Divide both sides by -10:
B = -1/10
7. Substitute the value of B back into equation 1 to solve for A:
A + B = 0
A + (-1/10) = 0
Add 1/10 to both sides:
A = 1/10
8. The partial fraction decomposition of 1/(x^2 - 25) is:
1/(x^2 - 25) = 1/10(x - 5) - 1/10(x + 5)
Therefore, the partial fraction decomposition of the given rational expression is 1/10(x - 5) - 1/10(x + 5).