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if (5x^(2) +2)/(x^(3)+x)=(A(1))/(x)+(A(...

if `(5x^(2) +2)/(x^(3)+x)=(A_(1))/(x)+(A_(2)x+A_(3))/(x^(2)+1), then (A_(1), A_(2), A_(3))=`

A

(0,2,3)

B

(2,0,3)

C

(3,0,2)

D

(2,3,0)

Text Solution

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The correct Answer is:
To solve the equation \[ \frac{5x^2 + 2}{x^3 + x} = \frac{A_1}{x} + \frac{A_2x + A_3}{x^2 + 1} \] we will follow these steps: ### Step 1: Rewrite the Denominator First, we can factor the denominator on the left side: \[ x^3 + x = x(x^2 + 1) \] Thus, we can rewrite the equation as: \[ \frac{5x^2 + 2}{x(x^2 + 1)} = \frac{A_1}{x} + \frac{A_2x + A_3}{x^2 + 1} \] ### Step 2: Combine the Right Side Next, we need to combine the right side into a single fraction: \[ \frac{A_1}{x} + \frac{A_2x + A_3}{x^2 + 1} = \frac{A_1(x^2 + 1) + (A_2x + A_3)x}{x(x^2 + 1)} \] This simplifies to: \[ \frac{A_1x^2 + A_1 + A_2x^2 + A_3x}{x(x^2 + 1)} = \frac{(A_1 + A_2)x^2 + A_3x + A_1}{x(x^2 + 1)} \] ### Step 3: Equate the Numerators Now we can equate the numerators of both sides: \[ 5x^2 + 2 = (A_1 + A_2)x^2 + A_3x + A_1 \] ### Step 4: Compare Coefficients Now we will compare the coefficients of \(x^2\), \(x\), and the constant term from both sides: 1. Coefficient of \(x^2\): \[ A_1 + A_2 = 5 \quad \text{(1)} \] 2. Coefficient of \(x\): \[ A_3 = 0 \quad \text{(2)} \] 3. Constant term: \[ A_1 = 2 \quad \text{(3)} \] ### Step 5: Solve the System of Equations From equation (3), we have: \[ A_1 = 2 \] Substituting \(A_1\) into equation (1): \[ 2 + A_2 = 5 \implies A_2 = 3 \] From equation (2), we already have: \[ A_3 = 0 \] ### Final Values Thus, we have: \[ (A_1, A_2, A_3) = (2, 3, 0) \] ### Conclusion The final answer is: \[ (A_1, A_2, A_3) = (2, 3, 0) \]
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