How to Express (3x 32) / (x - 46 - x) in Partial Fractions

Partial fractions is a technique in algebra that involves expressing a rational expression as a sum of simpler fractions. This method is particularly useful in integral calculus and simplifying complex fractions. This article will guide you through the process of decomposing the given rational function using partial fractions.

Solving the Rational Function (3x 32) / (x - 46 - x)

The given rational function is: (frac{3x 32}{x - 46 - x}). We will decompose this into simpler fractions using the partial fraction method. The general form for decomposition is:

(frac{3x 32}{x 46 - x} frac{A}{x - 4} frac{B}{6 - x})

Let's solve for the constants (A) and (B).

Step 1: Set Up the Equation

We start by multiplying both sides by the denominator (x 46 - x):

(3x 32 A(6 - x) B(x 4))

Step 2: Expand the Right Side

Expand and simplify the right side:

(3x 32 A(6 - x) B(x 4))

(3x 32 6A - Ax Bx 4B)

(3x 32 (B - A)x 6A 4B)

Step 3: Set Up a System of Equations

We now have a system of equations, equating the coefficients of (x) and the constant terms:

(B - A 3)

(6A 4B 32)

Step 4: Solve the System of Equations

Solve the first equation for (B) in terms of (A):

(B A 3)

Substitute (B A 3) into the second equation:

(6A 4(A 3) 32)

(6A 4A 12 32)

(10A 12 32)

(10A 20)

(A 2)

Now, substitute (A 2) back to find (B):

(B 2 3 5)

Step 5: Write the Partial Fraction Decomposition

Thus, the partial fraction decomposition of (frac{3x 32}{x - 46 - x}) is:

(frac{2}{x - 4} frac{5}{6 - x})

Final Answer: The partial fraction decomposition is (frac{2}{x - 4} frac{5}{6 - x}).

Alternative Methods for Verification

Alternatively, using the method of values:

1. Set (x -4) and factor out the denominator:

(frac{3x 32}{x - 46 - x} frac{A}{x - 4})

(3(-4) 32 A(6 - (-4)))

(A 2)

2. Set (x 6) and factor out the denominator:

(frac{3x 32}{x - 46 - x} frac{B}{6 - x})

(3(6) 32 B(6 - 6))

(B -5)

The partial fractions can also be determined directly:

(frac{3x 32}{x - 46 - x} A(x - 4) B(6 - x))

3x 32 x(6 - A) (4B - 4A))

Match coefficients:

(-A 3) and (6 - A 0)

(4B - 4A 32)

Solving, we find:

(A 2) and (B -5)

Therefore, the final partial fraction decomposition is: