Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{x^{5}-5 x^{4}+7 x^{3}-x^{2}-4 x+12}{x^{3}-3 x^{2}} $$

Step-by-Step Solution

Answer

\(x^2 - 2x + 1 + \frac{3}{x^2} - \frac{5}{x-3}\)

Step 1: Analyze the Degrees

The numerator is a polynomial of degree 5, and the denominator is a polynomial of degree 3. Since the degree of the numerator is greater than the degree of the denominator, we must first perform polynomial long division to simplify the expression.

Step 2: Perform Polynomial Long Division

Divide the numerator \(x^5 - 5x^4 + 7x^3 - x^2 - 4x + 12\) by the denominator \(x^3 - 3x^2\). The quotient is \(x^2 - 2x + 1\) and the remainder is \(-5x^2 + 4x + 12\). Thus, \(\frac{x^5 - 5x^4 + 7x^3 - x^2 - 4x + 12}{x^3 - 3x^2} = x^2 - 2x + 1 + \frac{-5x^2 + 4x + 12}{x^3 - 3x^2}\).

Step 3: Simplify the Denominator

Factor the denominator of the remaining fraction, \(x^3 - 3x^2\), as \(x^2(x - 3)\). This helps in setting up the partial fractions.

Step 4: Set Up Partial Fractions

Express \(\frac{-5x^2 + 4x + 12}{x^2(x-3)}\) as a sum of partial fractions: \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x-3}\).

Step 5: Clear the Denominator and Solve for Constants

Clear the denominators by multiplying through by \(x^2(x-3)\): \(-5x^2 + 4x + 12 = A x(x-3) + B(x-3) + Cx^2\). Plugging values or equating coefficients helps to find the constants: 1. Set \(x = 0\) to find \(B\). 2. Set \(x = 3\) to find \(C\). 3. Use another value or compare coefficients to find \(A\). Solve these equations to get \(A = 0\), \(B = 3\), \(C = -5\).

Step 6: Write the Partial Fraction Decomposition

Combine everything together: the original quotient and the constants found in the previous step. Hence, the decomposition is:\[x^2 - 2x + 1 + \frac{3}{x^2} - \frac{5}{x-3}\].

Key Concepts

Polynomial Long Division

When the degree of the numerator polynomial exceeds that of the denominator, polynomial long division becomes necessary. Imagine this process as similar to the long division you might perform with numbers, but here, we use polynomial terms. Begin by dividing the leading term of the numerator by the leading term of the denominator. This gives you the first term of the quotient.

• For example, divide the first term of the numerator, \(x^5\), by the first term of the denominator, \(x^3\), resulting in \(x^2\).
• Multiply the entire divisor by this first term of the quotient and subtract the result from the numerator.
• Repeat the process using the new polynomial obtained after subtraction as your new ‘numerator’ until the degree of what’s left (remainder) is lower than the degree of the original denominator.

This will leave you with a quotient and a remainder. The process ensures the original problem is broken down into a polynomial part plus a smaller fraction.

Numerator and Denominator Degrees

Understanding degrees is crucial for polynomial division and setting up partial fractions. The degree of a polynomial is the highest power of the variable in the polynomial expression.In our exercise:

• The numerator, \(x^5 - 5x^4 + 7x^3 - x^2 - 4x + 12\), has a degree of 5.
• The denominator, \(x^3 - 3x^2\), has a degree of 3.

Since the degree of the numerator is greater than the degree of the denominator, division is necessary before proceeding to partial fraction decomposition. This process ensures that any fraction leftover after division has a numerator with a lesser degree than its denominator. This step is critical because a fractional expression suitable for partial fraction decomposition requires the numerator degree to be less than the denominator's.

Factorization

Factoring the expression is a key step in simplifying the setup of partial fractions. After conducting polynomial long division, you'll typically encounter a remaining fraction that needs decomposing.In this exercise:

• The denominator \( x^3 - 3x^2 \) was factored into \( x^2(x - 3) \).

Factoring completely reveals fundamental building blocks of the polynomial, making it easier to set up partial fractions, as we can focus on each factor. Remember:

• If a polynomial factor is repeated, like \( x^2 \), it implies that in our partial fraction setup, terms like \( \frac{B}{x^2} \) could appear.
• Any distinct linear factors like \( x-3 \) will appear in the denominator of each partial fraction.

Setting Up Partial Fractions

After factoring the denominator, you set up the equation based on its factors to simplify the expression further. This is known as partial fraction decomposition.The objective is to express the fraction as a sum of simpler fractions:

• Each term in the denominator corresponds to a fraction in the decomposition.
• For our problem, we express the partial fractions as \( \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x-3} \).

Next, to solve for each unknown (\(A\), \(B\), and \(C\)), the denominators are cleared by multiplying through by the common denominator. You then solve the resulting equation by isolating terms, using substitute values (often values that make certain parts of the expressions zero), or comparing coefficients of like terms. This results in equations that allow resolving the constants, refining the partial fraction expression into a more understandable form.