Respuesta :
The partial fraction decomposition of the function is x⁴ + 2 / x⁵ + 4x³ = -1/8x + 1/2x³ + 9/8x(x²+4)
Partial fraction decomposition function:
In algebra, When an algebraic expression is split into a sum of two or more rational expressions, then each part is called a partial fraction.
Given,
Here we have the expression
x⁴ + 2 / x⁵ + 4x³
Here we need to find the partial fraction decomposition function.
In order to solve this, first we have to factorize the denominator, then we get,
=> x⁴ + 2 / x⁵ + 4x³ = (x⁴ + 2) / x³ (x² + 4)
Now, Partial fraction for each factors is written as,
=> A/x + B/x² + C/x³ + (Dx + E)/ (x² + 4)
Now, we have to Multiply through by the common denominator of x³ (x² + 4), then we get,
=> x⁴+2=A (x²(x²+4)) + B(x(x²+4)) + C (x²+4) + (Dx + E)(x³)
When we simplify this one, then we get,
=> x⁴ + 2 =Ax⁴ + 4Ax² + Bx³ + 4Bx + Cx² + 4C + Dx⁴ + Ex³
Group the x-terms and the constant terms
=> x⁴ + 2 =(A +D)x⁴ + (4A + C) x² + (B + E)x³ + 4Bx + 4C
Coefficients of the two polynomials must be equal, so we get equations
A+D=1
B+E=0
4A+C=0
4B=0
4C=2
After solving these equations, we get
A = -1/8, B = 0, C = 1/2, D=9/8, E = 0
Then when we substitute these values in the original fraction,
=> -1/8x + 1/2x³ + 9/8x(x²+4)
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