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Sagot :
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]
[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]
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