find the partial fraction decomposition of the equation.

Question

(4x^2-x-2)/(x^4+2x^3)

 

find the partial fraction decomposition of the equation.

Answer 1

x⁴+2x³ = x³(x+2)

 

(4x²-x-2)/(x⁴+2x³) = A/x + B/x²+C/x³+D/(x+2)

 

Multiply both sides by x³(x+2) to obtain:

 

4x²-x-2 = Ax²(x+2) + Bx(x+2)+C(x+2)+Dx³

 

4x²-x-2 = (A+D)x³ +(2A+B)x²+(2B+C)x+2C

 

Equating coefficients of like powers, we get:  A+D = 0

                                                                  2A+B = 4

                                                                  2B+C = -1

                                                                      2C = -2

From the last equation, C = -1

                  

So, 2B -1 = -1  Therefore, B = 0

 

Since 2A+B = 4 and B = 0, we get  A = 2

 

But, A+D = 0.  So, D = -2

 

The partial fraction decomposition is 2/x -1/x³ -2/(x+2)

Answer 2

Remember when you had two rational expressions that you had to add or subtract, and the result was a single rational expression.  With partial fraction decomposition, we are looking for the two expressions.

 

First, we need to factor the given expression using either FOIL or extracting the GCF.  The denominator part can be factored. This helps sets the stage in the process.

 

(4x² - x - 2) / [x³(x + 2)]

 

 

Now we can break this up into two parts.  So we have

 

(4x² - x - 2) / [x³(x + 2)] = [A / x³] + [B / (x + 2)]

 

where A and B are coefficients.

 

 

Note that on the right side of the equation, we are adding rational expressions with different denominators.  Therefore, we need to look for the LCD.  We know the LCD is x³(x + 2).

 

So we multiply the first rational expression by (x + 2) / (x + 2) and multiply the second rational expression by x³ / x³.

 

So the numerators on the right side of the equation is then

 

A(x + 2) + Bx³ = Ax + 2A + Bx³

 

 

Next, we equate the numerators on the left and right side of the equation.

 

4x² - x - 2 = Ax + 2A + Bx³

               

              

Next we equate coefficients.  This means that the coefficient of a type of variable on one side of the equation must equal the sum of the coefficients of the same type of variable on the other side of the equation.

 

x terms:  -1 = A

 

constant terms:  -2 = 2A

 

 

You have an x² term on the left side of the equation, but no x² terms on the right side of equation.  The same goes for the x³ term that is right side of the equation.  We only have equations to solve for A, but not B.   Because we have no equation for B, the two equations that are shown here are not valid.

 

Therefore, this entire question is not written correctly.  But the procedures that I have shown you here can be used to decompose a single rational expression.