Write the partial fraction decomposition of the rational expression.

Question

3/(x^2 + 5x + 4)

It would help me a lot if you could show all your work. Thank you!

Answer 1

You want to start out with factoring the polynomial on the bottom. Factoring would give you (x+4)(x+1). This means your new fraction is 3/(x+4)(x+1). Now, you want to separate this fraction into the sum of two fractions, with the numerators A and B (these are the two variables you want to solve for to find the partial fraction decomposition.

A/(x+4) + B/(x+1) = 3/(x+4)(x+1)

Cross multiplying gives you

(A(x+1) + B(x+4)) / (x+1)(x+4) = 3 / (x+4)(x+1)

Now that the denominators on both the sides are the same, you can just cancel them out, giving you

A(x+1) + B(x+4) = 3

Now let's start out by solving for A. In order to do this, we need to get rid of B. One way to do this is to set x equal to -4, so x+4 = 0. Multiplying B by 0 would get rid of B and only keep A in the equation.

A((-4) + 1) + B((-4) + 4) = 3

A(-3) = 3

A = -1

Now that we have A, we can do the same thing to derive B. Setting x to -1 would mean x+1 = 0, getting rid of A from the equation.

A (-1+1) + B((-1) +4) = 3

B(3) = 3

B = 1

Therefore the partial fraction decomposition for the fraction is

( -1 / (x+4) ) + ( 1 / (x+1) )

Answer 2

Hello Sam!

First, you have to separate the denominator as a multiplication of two expressions:

x² + 5x + 4 = (x+1)*(x+4)

The two fractions are:

A/(x+1) + B/(x+4)

Now looking at the numerator:

A*(x+4) + B*(x+1) = 3

A*x + B*x = 0

A = -B

4A + B = 3

-4B +B = 3

B = -1

A = 1

Therefore the fractions are:

1/(x+1) - 1/(x+4)

Feel free to schedule a class! Best regards!

Answer 3

First, need to factor the denominator

3/[(x+1)(x+4)]

Next, set up the equation.

A/(x+1)+B/(x+4)=3/[(x+1)(x+4)]

Multiply both side of the equation by the LCD to get rid of the fractions. The result is

(x+4)A+(x+1)B=3 Distribute

Ax+4A+Bx+B=3 Group the x terms and coefficient terms together.

Ax+Bx+4A+B=3 Next, split into 2 equations

A+B=0 (There is no x term on the right)

4A+B=3

This is 2 equations, 2 unknowns. Can solve by substitution or elimination. Multiply the first equation by -4

-4A+-4B=0

4A+B=3 Add them together

-3B=3 Divide both sides by -3

B=-1

Substitute in the first equation,

A=1

Final form is:

1/(x+1)+ -1/(x+4)