To perform partial fraction decomposition on the given rational expression:
[tex]\[
\frac{27}{x^2 - d^2}
\][/tex]
we start by recognizing that [tex]\(x^2 - d^2\)[/tex] can be factored using the difference of squares formula. Specifically:
[tex]\[
x^2 - d^2 = (x - d)(x + d)
\][/tex]
Therefore, we can rewrite the original expression as:
[tex]\[
\frac{27}{x^2 - d^2} = \frac{27}{(x - d)(x + d)}
\][/tex]
Next, we express the right-hand side as a sum of partial fractions. We assume:
[tex]\[
\frac{27}{(x - d)(x + d)} = \frac{A}{x - d} + \frac{B}{x + d}
\][/tex]
where [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are constants to be determined.
To find [tex]\(A\)[/tex] and [tex]\(B\)[/tex], we first combine the fractions on the right-hand side over a common denominator:
[tex]\[
\frac{A}{x - d} + \frac{B}{x + d} = \frac{A(x + d) + B(x - d)}{(x - d)(x + d)}
\][/tex]
Equate this to the left-hand side:
[tex]\[
\frac{27}{(x - d)(x + d)} = \frac{A(x + d) + B(x - d)}{(x - d)(x + d)}
\][/tex]
Since the denominators are the same, the numerators must also be equal:
[tex]\[
27 = A(x + d) + B(x - d)
\][/tex]
Now, let's solve for [tex]\(A\)[/tex] and [tex]\(B\)[/tex]. To do this, we will equate coefficients of corresponding powers of [tex]\(x\)[/tex] on both sides of the equation.
First, expand the right-hand side:
[tex]\[
27 = Ax + Ad + Bx - Bd
\][/tex]
Combine like terms:
[tex]\[
27 = (A + B)x + (Ad - Bd)
\][/tex]
For these to be equal for all [tex]\(x\)[/tex], the coefficients of [tex]\(x\)[/tex] and the constant term must match.
Comparing coefficients of [tex]\(x\)[/tex]:
[tex]\[
A + B = 0
\][/tex]
And comparing the constant terms:
[tex]\[
Ad - Bd = 27
\][/tex]
Since [tex]\(A + B = 0\)[/tex], we can solve for [tex]\(B\)[/tex] in terms of [tex]\(A\)[/tex]:
[tex]\[
B = -A
\][/tex]
Substitute [tex]\(B = -A\)[/tex] into [tex]\(Ad - Bd = 27\)[/tex]:
[tex]\[
Ad - (-A)d = 27
\][/tex]
This simplifies to:
[tex]\[
Ad + Ad = 27
\][/tex]
[tex]\[
2Ad = 27
\][/tex]
Solving for [tex]\(A\)[/tex]:
[tex]\[
A = \frac{27}{2d}
\][/tex]
Since [tex]\(B = -A\)[/tex]:
[tex]\[
B = -\frac{27}{2d}
\][/tex]
Now we can write the partial fraction decomposition:
[tex]\[
\frac{27}{(x - d)(x + d)} = \frac{27 / 2d}{x - d} + \frac{-27 / 2d}{x + d}
\][/tex]
[tex]\[
\frac{27}{(x - d)(x + d)} = \frac{27}{2d(x - d)} - \frac{27}{2d(x + d)}
\][/tex]
Thus, the partial fraction decomposition of the given rational expression is:
[tex]\[
\frac{27}{x^2 - d^2} = \frac{27}{2d(x - d)} - \frac{27}{2d(x + d)}
\][/tex]
