To calculate: The partial decomposition of x2−2x−21x3+7x

Question

Cherry McCormick · Accepted Answer

Calculation:
Consider the provided expression:

\frac{x^{2} - 2 x - 21}{x^{3} + 7 x}

Here,

f (x) = x ? - 2 x - 21 and g (x) = x^{\circ} + T x

Factorize

g (x) :

g (x) = x^{3} + 7 x

= x (x^{2} + 7)

So, the expression can be written as:

\frac{x^{2} - 2 x - 21}{x^{3} + 7 x} = \frac{x^{2} - 2 x - 21}{x (x^{3} + 7 x)}

Here, in g (x), there are two factors, one linear and one quadratic. So, the expression can be decomposed as:

\frac{x^{2} - 2 x - 21}{x (x^{3} + 7 x)} = \frac{A}{x} + \frac{B x + C}{x^{2} + 7}

Here, Least Common Divisor is

x (x^{2} + 7)

Multiply both sides of (1) by the Least Common Divisor to clear fractions:

x (x^{2} + 7) [\frac{x^{2} - 2 x - 21}{x (x^{2} + 7 x)}] = x (x^{2} + 7) [\frac{- A}{x} + \frac{B x + C}{x^{2} + 7 x}]

simplify to obtain:

x^{2} + 2 x - 21 = A (x^{2} + 7) + (B x + C) x

x^{2} + 2 x - 21 = A x^{2} + 7 + B x^{2} + C x

x^{2} + 2 x - 21 = (A + C) x^{2} + C x + 7 A

Compare the coefficients of x,x? and constant terms:

A + B = 1, C = - 2 nd7A=-2 n d 7 A = - 21

Solve for A the equation

7 A = — 21

7 A = - 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):

\frac{x^{2} - 2 x - 21}{x (x^{3} + 7 x)} = \frac{3}{x} + \frac{4 x - 2}{}

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