To calculate: The partial decomposition of \frac{x^{2}-2x-21}{x^{3}

goymdujf

goymdujf

Answered question

2021-11-18

To calculate: The partial decomposition of x22x21x3+7x

Answer & Explanation

Cherry McCormick

Cherry McCormick

Beginner2021-11-19Added 23 answers

Calculation:
Consider the provided expression:
x22x21x3+7x
Here, f(x)=x?2x21andg(x)=x+Tx
Factorize g(x):
g(x)=x3+7x
=x(x2+7)
So, the expression can be written as:
x22x21x3+7x=x22x21x(x3+7x)
Here, in g (x), there are two factors, one linear and one quadratic. So, the expression can be decomposed as:
x22x21x(x3+7x)=Ax+Bx+Cx2+7
Here, Least Common Divisor is x(x2+7)
Multiply both sides of (1) by the Least Common Divisor to clear fractions:
x(x2+7)[x22x21x(x2+7x)]=x(x2+7)[Ax+Bx+Cx2+7x]
simplify to obtain:
x2+2x21=A(x2+7)+(Bx+C)x
x2+2x21=Ax2+7+Bx2+Cx
x2+2x21=(A+C)x2+Cx+7A
Compare the coefficients of x,x? and constant terms:
A+B=1,C=2nd7A=-2nd7A=21
Solve for A the equation 7A=21
7A=21
A=3
Substitute -3 for A in A+B=1 and simplify for B:
(3)+B=1
B=4
Substitute the obtained values of A, B and Cin (1):
x22x21x(x3+7x)=3x+4x2

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