Question

Find the partial fraction decomposition of $$ \left(3 x^{3}-12 x^{2}+21-3\right) /\left\\{(x+1)(x-2)^{3}\right\\} $$

Short answer

The partial fraction decomposition of the given rational function is: \[ \frac{3x^3 -12x^2 + 21 - 3}{(x + 1)(x - 2)^3} = 3 - \frac{2}{x + 1} + \frac{4}{(x - 2)^2} + \frac{6}{(x - 2)^3} \]

Step-by-step solution

Analyze the degree of numerator and denominator

The given rational function is \[ \frac{3x^3 -12x^2 + 21 - 3}{(x + 1)(x - 2)^3} \] The degree of the numerator is 3 and the denominator is also 3 (since there's a factor of \((x-2)^3\)). The degrees are equal, so we can proceed with polynomial long division.

Polynomial long division

Perform polynomial long division between the numerator and the denominator: Dividend = \(3x^3 -12x^2 + 21 - 3\) Divisor = \((x+1)(x-2)^3 = x^3 - 6x^2 + 12x - 8\) Performing the long division, we get: Quotient (Q) = \(3\) and Remainder (R) = \(6x^2 -12x + 48\) So now, our rational function becomes: \[ \frac{3x^3 -12x^2 + 21 - 3}{(x + 1)(x - 2)^3} = 3 + \frac{6x^2 -12x + 48}{(x + 1)(x - 2)^3} \]

Express as partial fraction decomposition

Now, let's express the remainder as a sum of partial fractions: \[ \frac{6x^2 -12x + 48}{(x + 1)(x - 2)^3} = \frac{A}{x + 1} + \frac{B}{x - 2} + \frac{C}{(x - 2)^2} + \frac{D}{(x - 2)^3} \] We need to find the values of A, B, C, and D.

Clear the denominators and find the coefficients

To find A, B, C, and D, we will clear the denominators by multiplying both sides by the common denominator \((x+1)(x-2)^3\): \[ 6x^2 -12x + 48 = A(x-2)^3 + B(x+1)(x-2)^2 + C(x+1)(x-2) + D(x+1) \] Now we can find the coefficients by solving for specific values of x: 1. For \(x = -1\), we get: \[ 6 + 12 + 48 = -27A \Rightarrow A = -2 \] 2. For \(x= 2\), we get: \[ 24 - 24 + 48 = 8D \Rightarrow D = 6 \] Now, differentiate both sides of the equation with respect to x and substitute the values of A and D: \[ 12x - 12 = -6(x-2)^2 + 2B(x-2)(x+1) + C(x+1) + 6 \Rightarrow (1) \] 3. By substituting \(x = 2\) in equation (1), we get: \[ 12 = C(3) \Rightarrow C = 4 \] 4. Now, we have A, C, and D. We can substitute them back into equation (1) and solve for B: \[ 12x - 12 = -6(x-2)^2 + 2B(x-2)(x+1) + 4(x+1) + 6 \] Setting \(x=-1\), we get: \[ 0= 2B(-1-2)(-1+1) \Rightarrow B=0 \]

Write the final partial fraction decomposition

Now that we have all the coefficients, we can write the final partial fraction decomposition: \[ \frac{3x^3 -12x^2 + 21 - 3}{(x + 1)(x - 2)^3} = 3 - \frac{2}{x + 1} + \frac{4}{(x - 2)^2} + \frac{6}{(x - 2)^3} \] So the partial fraction decomposition of the given rational function is: \[ 3 - \frac{2}{x + 1} + \frac{4}{(x - 2)^2} + \frac{6}{(x - 2)^3} \]

Key concepts

Polynomial Long Division

Polynomial long division is similar to traditional long division but involves dividing polynomials instead of numbers. It's a method used to simplify expressions where the degree of the numerator is greater than or equal to the degree of the denominator.

This process enables us to rewrite a rational function as a sum of a polynomial and a simpler rational function. Our example has a numerator of degree 3 and a denominator that is also equivalent to degree 3 because of \( (x - 2)^3 \).

Here’s a brief rundown of the steps:

• Divide the leading term of the numerator by the leading term of the denominator to find the first term of the quotient.
• Multiply the entire divisor by this new term and subtract from the dividend.
• Repeat the process with the new (reduced) polynomial until no remaining terms can be divided by the divisor.

This results in a quotient, here represented by the number 3, and a remainder, which is expressed for further analysis.

Rational Functions

Rational functions are expressions that are ratios of two polynomials. The form of a rational function is \(\frac{P(x)}{Q(x)}\), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \) is not equal to zero.

In the context of partial fraction decomposition, rational functions are decomposed into simpler fractions. These simpler components are easier to work with, especially in calculus for integration or in developing series expansions.

Our specific rational function example is \( \frac{3x^3 - 12x^2 + 21 - 3}{(x + 1)(x - 2)^3} \), which is a classic setup for decomposition because its denominator can be factored.

This function was simplified using polynomial long division, leading to a more manageable expression that could then be broken into parts using partial fractions.

Degree of Polynomials

The degree of a polynomial is defined as the highest power of the variable in its terms. For instance, in the polynomial \( 3x^3 - 12x^2 + 21 - 3 \), the highest power of \( x \) is 3, thus the degree is 3.

Understanding polynomial degree is crucial when comparing the degrees of the numerator and denominator of a rational function. This comparison helps determine which mathematical techniques, such as polynomial long division, might be necessary.

In our problem, since both the numerator and the factored denominator have the same degree (3), we utilized polynomial long division to process the equation. This provided us with a quotient and a new lower-degree polynomial to further analyze with partial fraction decomposition.

Coefficients in Algebraic Expressions

Coefficients in algebraic expressions are the numerical factors multiplied by variables. They tell us 'how much' of a variable is present in a term.

In the polynomial \( 3x^3 - 12x^2 + 21 - 3 \), note the coefficients:

• 3 is the coefficient of \( x^3 \)
• -12 is the coefficient of \( x^2 \)
• 21 is a constant term (or you can think of it as having a coefficient of 21 for \( x^0 \))

In our partial fraction decomposition, identifying coefficients correctly allows us to utilize algebra to solve for unknowns like \( A, B, C, \) and \( D \). These coefficients were chosen to make both sides of the equation equivalent.

By strategically selecting values for \( x \) and solving these equations, we could isolate and calculate the coefficients for each term in the partial fraction decomposition, composing a complete and correct solution.