Answer:
To resolve the fraction [tex]\frac{x^2+2x}{(x+3)(x^2 + 1)}[/tex] into partial fractions, we can follow these steps:
Step 1: Factorize the denominator.
The denominator is given by (x+3)(x^2 + 1).
Step 2: Write the fraction in the form of partial fractions.
The fraction can be expressed as:
[tex]\frac{x^2+2x}{(x+3)(x^2 + 1)} = \frac{A}{x+3} + \frac{Bx + C}{x^2 + 1}[/tex]
Step 3: Find the values of A, B, and C.
To find the values of A, B, and C, we can multiply both sides of the equation by the common denominator, which gives us:
[tex]x^2+2x = A(x^2 + 1) + (Bx + C)(x + 3)[/tex]
Expanding and equating the coefficients of like powers of x:
For the constant term: 0 = 3C + A
For the coefficient of x: 2 = B + 3C
For the coefficient of x^2: 1 = A
From equation 3, we get A = 1.
Substituting A = 1 in equations 1 and 2:
From equation 2, we get 2 = B + 3C.
From equation 1, we get 0 = 3C + 1.
Solving these equations, we find C = -
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Answers & Comments
Answer:
To resolve the fraction [tex]\frac{x^2+2x}{(x+3)(x^2 + 1)}[/tex] into partial fractions, we can follow these steps:
Step 1: Factorize the denominator.
The denominator is given by (x+3)(x^2 + 1).
Step 2: Write the fraction in the form of partial fractions.
The fraction can be expressed as:
[tex]\frac{x^2+2x}{(x+3)(x^2 + 1)} = \frac{A}{x+3} + \frac{Bx + C}{x^2 + 1}[/tex]
Step 3: Find the values of A, B, and C.
To find the values of A, B, and C, we can multiply both sides of the equation by the common denominator, which gives us:
[tex]x^2+2x = A(x^2 + 1) + (Bx + C)(x + 3)[/tex]
Expanding and equating the coefficients of like powers of x:
For the constant term: 0 = 3C + A
For the coefficient of x: 2 = B + 3C
For the coefficient of x^2: 1 = A
From equation 3, we get A = 1.
Substituting A = 1 in equations 1 and 2:
From equation 2, we get 2 = B + 3C.
From equation 1, we get 0 = 3C + 1.
Solving these equations, we find C = -
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